Science And Technology » Physics http://oscience.info The ultimate resource for Science and Technology Tue, 08 May 2012 16:59:04 +0000 en hourly 1 http://wordpress.org/?v=3.3.2 Elecric Field http://oscience.info/physics/electric-field/elecric-field/ http://oscience.info/physics/electric-field/elecric-field/#comments Thu, 05 Apr 2012 08:13:34 +0000 sandeep http://oscience.info/?p=6711 Frictional Electricity:

Frictional electricity is the electricity caused due to friction. When two neutral bodies (e.g. glass rod and silk) are rubbed together, both gets charged and the charges produced on them are opposite in nature. The reason of both bodies being electrified is that some of electrons of one body are transferred to the other.
Types of charges: Charges are of two types:

(i) Positive charges

(ii) Negative charges

Electrons possess negative charge. Therefore when a body gains electrons, it becomes negatively charged but when it gives off electrons, it becomes positively charged.
Thus positively charge means deficiency of electrons and negative charge means excess of electrons.
Interaction between charged bodies:
It is found experimentally

(i) That the similarly charged bodies repel; while oppositely charged bodies attract. In other words similar charges repel while opposite charges attract.

(ii) A charged body also attracts a light neutral body.

Thus if a body (A) is attracted by a charged (or electrified) body; then the body (A) may be oppositely charged or neutral.
This implies that attraction is not the sure test of electrification of body.
On the other hand, if a body is repelled by a charged body; it must be similarly charged. This implies that the repulsion is the sure test of electrification.

Conservation of charge:

The net charge on a system remains constant. This means that the charge can neither be created nor destroyed, but it may merely be transferred from one body to another. When the two objects are rubbed, the electrons of one object are transferred to the other.

Unit of charge:

In S.I. system the unit of charge is Coulomb abbreviated as C.
In C.G.S. system the unit of charge is often written as e.s.u. of charge or stat-coulomb.

1 coul = 3 x 109 stat-coul
In practice coulomb is big unit; therefore small units microcoulomb is often used.
1 microcoulomb (\mu C = 10^{-6})

Dimensions of charge Q are [Q]: [AT].
Elementary Unit : As charge is produced due to transfer of electrons; the magnitude of charge on electron is taken as elementary unit of charge (or elementary charge or, fundamental charge).
1 elementary unit, e = charge on electron

= 1.6 x 10-19 Coulomb

Quantum Nature of charge or principle oi atomicity of charge :

 A physically existing charge Q is always on integral multiple of elementary charge (e) i.e. Q = ne.
where n may be positive or negative integer.
Accordingly charge can take values e, 2e, 3e: ——-, but it cannot take values 1.5e, 2.2e, 3.8e —-~ etc.
This is called principle of quantization (or atomicity) of charge.

If charge is 1 coulomb, then

n = \dfrac{Q}{e} = \dfrac{1 coul}{1.6 \times 10^{-19} coul} = 6.25 \times 10^18

Thus, 1 coulomb charge is equivalent to deficiency of 6.25 x 1018 electrons

Coulomb’s Law:

The force of attraction or repulsion between two point charges (q1 and q2) at
finite separation (r) is directly proportional to the product of charges and inversely proportional to the square of distance between the charges i.e.

F_m \propto \dfrac{q1 q2}{r^2} \text{or} F = \dfrac{1}{4 \pi \varepsilon} . \dfrac{q1 q2}{r^2} \dots (1)

where \varepsilon is the permittivity of medium between the charges.
If charges are separated by free space (or air), then \varepsilonis replaced by \varepsilon _0, where \varepsilon _0 is the permittivity of free space (or air); so

\text{force} F = \dfrac{1}{4 \varepsilon _0}. \dfrac{q_1q_2}{r^2} \cdots (2) \\ \text{The permittivity of free space } \\ \varepsilon _0 = 8.86 \times 10^{-12} {coul} ^2 / N- m ^2 \\ \text{and} \\ \dfrac{1}{4 \pi \varepsilon _0} = \dfrac{1}{4 \times 3.14 \times 8.86 \times 10^{-12}} \\ = 9 \times 10^9 N- m^2 \text{coul}^2

Dielectric constant:

The ratio of permittivity of medium ( \varepsilon ) to the permittivity of free space (\varepsilon _0) is called the dielectric constant or relative permittivity of medium

Dielectric constant K = \dfrac{\varepsilon}{\varepsilon _0} \dots (3)

Alternatively the dielectric constant of a  medium is the ratio of force between two charged particles in air and in medium

i.e.     K = \dfrac{F_{\text{air}}}{F_{\text{medium}}} \dots (4)

Equation (1) may also be expressed as

F = \dfrac{1}{4 \pi \varepsilon} . \dfrac{q1 q2}{r^2} = \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{q1 q2}{K r^2} \dots (5)

Remarks: 1.

The force is a vector quantity, while calculating the force from Coulomb’s law, the sign of charge is not retained in formula; but the signs of charges indicate direction of force which is seen by inspection with the rule that the charge on which force is to be calculated is assumed to have the tendency of motion while the other charge due to which force is to be calculated is assumed at rest, unless otherwise stated.
If the force on a charge qo due to a number of charges to be found, the forces on charge due to individual charges are calculated and then summed vectorially to find the resultant.

Electric Field Lines of Force:

The electric field is a region in which a charged particle experiences a force.
The electric field strength ( \overrightarrow{E} ) at any point is defined as the force experienced by per unit infinitesimal positive charge when placed at that point i.e.

\overrightarrow{E} = \lim_{q_0 \to 0} \frac{\overrightarrow{F}}{q_0} \dots (6)

where qo is infinitesimal positive test charge.
The electric field strength is a vector quantity. Its, direction is away from a positive charge and towards the negative charge. Its unit is Newton coulomb or volt/meter.
Michael Faraday always thought a vector field in terms of lines of force. The lines of force form a convenient way of visualizing electric field patterns. The magnitude of electric field strength at any point is measured by the number of electric lines of force passing per unit small area around that point normally and the direction of field at any point is given by the tangent to the line of force at the point.

Properties of lines of force :
1. The lines of force diverge out from a positive charge and converge at a negative charge. More correctly the lines of force are always directed from higher to lower potential.
2. The electric lines of force contract lengthwise and expand laterally.
3. Two lines of force never intersect. If they are assumed to intersect, there will be two directions of electric field at the point of intersection: which is impossible.

 Calculation of electric field strength

Electric field due to point charge:

Let q be a point charge producing electric field. The force on a positive test charge qo at separation r from charge q is

F = \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{q q_0}{r^2} \\ \therefore \, \, \text{Field strength } E = \dfrac{F}{q_0} = \dfrac{1}{4 \pi \varepsilon _0} .\dfrac{q}{r^2} \dots (7)

Electric Field due to a group of charges:

To find the field due to a group of point charges q1, q2,…..qn, the field \overrightarrow{E1}, \overrightarrow{E1}, \dots \overrightarrow{En} at given point due to individual point charges are found and then vectorially added to find the resultant field \overrightarrow{E1}  i.e

\overrightarrow{E} = \overrightarrow{E1} + \overrightarrow{E1} + \dots \overrightarrow{En} \dots (8)

Electric Flux:

The total number of electric lines of force through a given area normally is called the electric flux. The electric flux through a surface element \triangle \overrightarrow{S} is

\triangle \phi = \overrightarrow{E} . \triangle \overrightarrow{S}

Net electric flux through the whole surface S is

\phi = \sum E . \triangle \overrightarrow{S} = \int \overrightarrow{E}. \overrightarrow{dS} \, \, \, \, ( \text{a scalar } )

Gauss’s Theorem:

The net electric flux through a closed surface is equal to 1 / \varepsilon _0times the net charge within the surface . i.e.

\phi = \dfrac{1}{\varepsilon _0} \times \text{charge enclosed by surface}

Mathematically \int_s \overrightarrow{E} . \overrightarrow{dS} = \dfrac{1}{\varepsilon _0} \times Q
where Q is net charge enclosed by the surface.

Electric potential:

The electric potential at any point in an electric field is defined as the work done required to move per unit small positive test charge from infinity to that point. If W is the work done in bringing the charge q0 from infinity to given point P, then potential at P

V = \lim_{q_0 \to 0} \frac{W}{q_0}

Alternatively the electric field at any point in an electric field is defined as the negative line integral for the electric field vector \overrightarrow{E} from a infinitely away from all charges (giving rise to ( \overrightarrow{E}), to that point i.e.

V =- \int_{\infty}^{r} \overrightarrow{E} . \overrightarrow{dr}

The potential difference between two points is the work done in bringing per unit small positive change from one point to another. The unit of electric potential is joule/coul or volt.

The electric potential due to a point charge q at separation r is

V = \dfrac{1}{4 \pi \varepsilon _0 }. \dfrac{q}{r} ( \text{sign of charge is to be retained} )

The electric potential due to a group of charges q1,q2,….qn is

V = V1 + V2 + V3 + …..+ VN

\dfrac{1}{4 \pi \varepsilon} . ( \dfrac{q_1}{r_1} + \dfrac{q_2}{r_2} + \dfrac{q_n}{r_n} = \dfrac{1}{4 \pi \varepsilon} \sum \dfrac{q_i}{r_i}

where r I ; is the distance of point from charge q i.
The relation between electric field and potential is

E = - \dfrac{\triangle V}{\triangle r} = \dfrac{V}{d} ( \text{numerically} )

Electric potential energy:

The electric potential energy of a system of point charges is defined as the work sane in assembling this system of charges from an infinite distance apart from one another: It is assumed that all the charges are at test when they are at infinite distance apart from one another (i.e. their initial kinetic energy is zero).
The electric potential energy of two point charges q1 and q2 at separation r12 is

U_12 = \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{q_1q_2}{r_{12}} \dots (1)

In general for a system of n charges, the electric potential energy is

U = \dfrac{1}{4 \pi \varepsilon _0} \sum_{j = 1}^n \sum_{j>1}^n \dfrac{q_i q_j}{r_{ij}}

Electron volt:

A small unit of energy: When a charge q is accelerated through a potential difference of V volts, the energy gained by charge = qV.
When one electronic charge (= 1.6 X 10-19 coul) is accelerated through a potential difference of one volt, then energy is called 1 electron volt. Electron volt abbreviated as eV is a small unit of energy in Particle Physics.

1 ev = 1.6 X 10-19 coul

Electric dipole:

Two equal and opposite charges separated by a finite distance constitute an electric dipole. If –q and +q are charges at distance 2l apart, then dipole moment

\overrightarrow{p} = \overrightarrow{q} . 2 \overrightarrow{l} \dots (1)

An electric dipole in a uniform field:

When an electric dipole is placed in a uniform electric field E, at an angle \theta with the direction of field, the charges + q and –q  experience forces qE and qE in opposite directions, so that force on dipole = 0.

Its direction is directed from –q to +q. The torque on a dipole in uniform electric field

t = q E ( 2l \sin \theta ) = ( q . 2l ) E \sin \theta \dots (1) \\ pE \sin \theta

Remark: If the electric field is non-uniform, then electric dipole placed in the field experiences both force and torque.
Potential energy of electric dipole, U = -pE \cos \theta
Work done in rotating the dipole from angle \theta _1 to \theta _2; is equal to the difference of potential energy of dipole is two orientations.

i.e. U = U_2 - U_1 = pE \cos \theta _2 - ( - p E \cos \theta _1 ) \\ pE ( \cos \theta _1 - \cos \theta _2 )

Potential due to a short dipole:

(i) At the axis : On a distance r from dipole

The electric field E = \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{2p}{r^3}

The potential V = \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{p}{r^2}

(ii) At the equator: At a distance r from dipole

The electric field E = E = \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{p}{r^3} parallel to dipole along +q to –q.
The potential V = 0.

Electric potential at general point P (r, \theta );
due to short dipole is

Electric field and potential in some special cases:

#Due to a charged spherical conductor of charge Q, radius R at distance r:

Field outside point , E ( r > R ) = E = \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{Q}{r^2}

Surface point, E (r = R) = \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{Q}{R^2}

Interior point , E ( r < R ) = 0

Potential

Outside point , V ( r > R ) = \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{Q}{R}

Potential at interior point is same as on the surface
i.e. V ( r = R ) = V ( r < R ) = \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{Q}{R}

#Due to a uniformly non-conducting charged of sphere of charge Q and radius R at distance r
Field.

Outside point, E ( r > R ) =  \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{Q}{r^2}

Surface point, E ( r = R ) =  \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{Q}{R^2}
Interior point, E (r < R) =  \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{Q r}{R^3}
Potential

V ( r > R ) =  \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{Q}{r}

V( r = R ) =  \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{Q}{R}

V ( r < R ) =  \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{Q ( 3 R^2 - r^2 )}{2R^3}

# For a long line charge of linear charge density q, electric field at separation r,

E = \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{2q}{r}

Potential between two points at distance r1 and r2 from the wire is

V  =  \dfrac{1}{4 \pi \varepsilon _0} 2q \log _e \dfrac{r_2}{r^1}

# Near a thin flat sheet of charge of surface charge density \sigma \\ E = \sigma / 2 \varepsilon _0,\.

#Near a conductor of any shape, E = \sigma / \varepsilon _0

#The electric field at the axis of a uniformly charged ring of charge q, and radius R at a distance x from centre is

E = \dfrac{1}{4 \pi \varepsilon _0} . \dfrac{qx}{( R^2 + x^2 )^{\dfrac{3}{2}}}

Kinetic energy of a charged particle accelerated through a potential difference of V volts is

EK = qV

If v is the velocity gained, then

½ mv2 = qV

or v = \sqrt{2 \dfrac{qV}{m}}

Path of charged particle in Transverse Electric Field:

Let the electric field E be along Y-axis and the charged particle q, mass m enter the field along X-axis with speed u. The electric force on the particle is along Y-axis given by

The initial velocity of the particle is u along X-axis and it is zero along Y-axis. Therefore the deflection of charged particle along Y-axis after time is

y = uy t + ½ ay r 2

= 0 + \dfrac{1}{2} . \dfrac{qE}{m}r^2 = \dfrac{1}{2} . \dfrac{qE}{m}r^2 \dots (3)

As there is no acceleration along X-axis, therefore the distance traversed by particle in time t along X—axis is
x = u t …..(4)

Eliminating from (3) and (4), we get

y = \dfrac{1}{2} . \dfrac{qE}{m} ( \dfrac{x}{u} )^2 i.e. y = \dfrac{qE}{2mu^2} x^2

This shows that the path of charged particle in transverse electric field is a parabola.

Energy associated with Electric Field:

The electric energy stored in electrified per unit volume.

u_e = \dfrac{1}{2} \varepsilon E^2
This is called electric energy density.

]]> http://oscience.info/physics/electric-field/elecric-field/feed/ 0 Doppler’s effect http://oscience.info/physics/dopplers-effect/dopplers-effect/ http://oscience.info/physics/dopplers-effect/dopplers-effect/#comments Thu, 05 Apr 2012 07:01:02 +0000 sandeep http://oscience.info/?p=6705 Doppler’s Effect:

The apparent change in frequency of a sounding body due to relative motion between source and observer is called Doppler’s effect. Doppler’s effect takes place both in sound and light. In sound it depends on whether the source or observer or both are in motion; while in light it depends only on the fact that whether the distance between source and observer is decreasing or increasing.

Doppler’s Effect in Sound

(i) Source in motion and observer at rest.

If source is approaching the observer with speed vs, then n waves emitted per second are spread in distance v – vs instead of v.
Therefore apparent wavelength

\lambda' = \dfrac{v - v_s}{n}

or the apparent frequency n’ is given by

n' = \dfrac{v}{\lambda'} = \dfrac{v}{v - v_s} n

Similarly, if source is moving away from the observer; then

n' = \dfrac{v}{v + v_s} n

(ii) Source at rest and observer in motion.

If observer is moving away from the source with speed vo, then the observer does not receive the waves in distance v0, therefore the apparent frequency

n' = n - \triangle n \dfrac{v_0}{\lambda} = n - \dfrac{v_0}{v/n} \\ = ( 1 - \dfrac{v_0}{v} ) n = \dfrac{v - v_0}{v} n

Similarly, If observer its approaching the source, then apparent frequency

n' = n + \triangle n = \dfrac{v + v_0}{v} n

(iii) Source and observer are both in motion.

When source and observer are both in motion along the same direction, then apparent frequency due to motion of source alone is increased to

while that due to motion of observer is decreased to

n' = \dfrac{v - v_0}{v} n_1 = \dfrac{v - v_0}{v} . \dfrac{v}{v - v_s} n = \dfrac{v - v_0}{v - v_s} n

Remark :

(i) If wind also blows with speed \omega then v should be replaced by v + \omega \cos \theta where \theta is the angle between direction of propagation of sound and direction of wind. If wind blows along the direction of propagation of sound  \theta = 0 or \cos \theta = 1 but if it blows in opposite direction \theta = 180o or \cos \theta = - 1

(ii) In the solution of problems the formula for apparent frequency is chosen as

n' = \dfrac{v - v_0}{v - v_s} n

If wind blows with velocity \omega in a direction at angle \theta with \overrightarrow{v} \\ n' = \dfrac{v + \omega \cos \theta - v_0}{v + \omega \cos \theta - v_s}

If direction of source or observer changes, the formula is modified taking proper sign for velocities.

Doppler’s Effect in Light.

If v is actual frequency of a light source and v’ the apparent frequency then for approach

v' = \sqrt{1 + v / c}{1 - v / c} v \, i.e. v' > v \\ \text{and} \triangle v = v' - v = \dfrac{v}{c} v if v << c ,
v being relative velocity of source and observer and c = speed of light in vacuum = 3 x 108 m/s

 For receding

v' = \sqrt{ ( \dfrac{1 + v/c}{1 - v/c} ) } v \, i.e. v' <c \\ \text{and} \triangle v = v - v' = \dfrac{v}{c} \lambda \text{and} \lambda' \text{are actual and observed wavelength , then} \\ \triangle \lambda = \lambda - \lambda' = \dfrac{v}{c} \lambda \text{for approval} \\ \triangle \lambda = \lambda' - \lambda = \dfrac{v}{c} \lambda \text{for recording}

Characteristics of Musical Sound.

There are three characteristics of sound :

(i) Loudness or intensity:

The intensity of sound is defined as the amount of sound energy crossing per unit area around a point per second. The intensity I is given by

I = 2 \pi ^2 n^2 a^2 \rho v \dots (1)

when   n=frequency of wave, a = amplitude, \rho= density of medium, v = speed of wave
The level of sound is measured in bels and decibels.

Number of bels = \log_10 \dfrac{I}{I_0}

Number of decibels = 10 \log_10 \dfrac{I}{I_0}
where I0 is reference intensity = 10-12 watt/m2.

The loudness of sound is the degree of sensation produced in the ear and it depends on intensity  as well as upon the sensitiveness of ear. The loudness
L and intensity I are related as

L = k log 1                                   …(2)

(ii) Pitch or Frequency:

The pitch is the characteristic which distinguishes between a shrill (or sharp) sound and a grave (or flat) sound. A sound of high pitch is said to be shrill and of low pitch a grave sound. The pitch does not depend on intensity and loudness but depends on the frequency. The pitch of a sound changes due to Doppler’s effect.

(iii) Quality or Timbre:

The quality of sound enables us to distinguish between two sounds having same loudness and pitch. The quality of sound depends on the presence of overtones. Due to quality of of sound one can recognise the voice of his friend without seeing him.

 

Musical Scale:

A series of notes sounded in succession in such a way that their frequencies have a definite ratios and which produce a pleasant on the ear is called a musical scale.
The ratio between two frequencies of two notes of musical scale is called a musical interval.

Major Diatonic Scale: The most common musical scale is Diatonic Scale. It consists of eight
notes in ascending order of frequencies, comprising – an octave i.e. the frequency ratio of eighth and first note is 2 : 1. Conventionally the fundamental frequency of first note is taken as 256 Hz and that of the last note as 512 Hz. The following table illustrates the frequencies of intermediate notes along with their western and Indian names.


Clearly the frequency ratio of adjacent notes in diatonic scale bear simple ratios i.e. either 9/8 or 10/9 or 16/15.

Key Note:

The note C of lowest frequency is called the key note and is usually taken to be of frequency 256 Hz or 28 Hz.

Major Tone:

The interval between two notes having frequency ratio 9/8 is called major tone.

Minor Tone:

The interval between two notes having frequency ratio 10/9 is called a minor tone.

Half Tone:

The interval between two notes having frequency ratio 16/15 is called a half tone.
The major diatonic scale has the following characteristics.
(i) The interval between any note of the diatonic scale with the key note bears a simple ratio.
(ii) The interval between any two adjacent notes of the diatomic scale also bears a simple ratio.
(iii) The interval between any two notes is the equal to the sum (in musical scale the product of intervals is called the sum) of the intervals between the intermediate notes.
For example take intervals E and B.
The interval between E and B = \dfrac{480}{320} = \dfrac{3}{2}
The sum of intermediate intervals is \dfrac{16}{5} \times \dfrac{9}{8} \times \dfrac{10}{9} \times \dfrac{9}{8} \times \dfrac{3}{2}

Practical Difficulty in Diatonic Scale:

The first note of diatonic scale is called the key note (or tonic) note of the scale. In diatonic scale C is key note. The characteristic of music is that a singer often changes his key (or tones) note he may start with C or D or E as key note. If he starts with D or E, then another scale with above given intervals has to be constructed, otherwise the sound will appear unpleasant. To remove the difficulty, equally tempered scale has been constructed.

Equally Tempered Scale:

It consists of 13 keys and 12 intervals. The intervals are equal and
each interval is 2 1/2.
In this scale few new notes have been introduced. They are Cs, Ds, Fs, Gs and As. The main advantage of this scale in that the interval is same between the consecutive notes and so a singer can conveniently use any key as his tonic. Clearly the intervals in equally tempered scale form a geometric progression. In the Harmonium 13 keys are provided. 8 white keys and 5 black keys (fig.) 8 white keys

represent appropriately the diatonic scale. If a singer starts with C as key note, he can play only with white keys forming diatonic scale, but if he uses any other key’ as key note, black keys have to be used. Usually 3 or 4 octaves are provided in harmonium or piano.
4. Acoustics of Buildings.
Acoustic of buildings deals with the design and construction of halls, auditoria etc. with respect to sound waves. If the speech and music delivered in a hall or an auditorium is audible to listeners at all positions without any disturbance, the hall or auditorium is said to be accoustically good.
W.C. Sabine in 1911, studied the problem and laid down the following essential features or demand for a hall or auditorium for its being acoustically good:

(i) The sound must be large enough at each position of the hall.
(ii) There should be no echo so that the quality of sound should remain unchanged.
(iii) The successive syllables spoken should be clear without any echo.
(iv)There should be no noise at all.

Let us now consider each feature one by one.

Sufficient loudness:

At each position of the hall, sufficient loudness is essential. Reflecting surfaces like walls and ceiling of the hall are quite helpful in providing proper loudness.
The walls-behind the speaker and ceiling -reflect the sound of speaker and help in increasing the loudness. The reflection of sound to the back is very useful in increasing the sound intensity there. Therefore, it is useful to make back and side walls in the vicinity of the speaker in the form of reflector with the speaker at the focus. By this method the sound may be heard uniformly in the hall by a large gathering, but this method has following two shortcomings.
(i) The area of speaker becomes-limited.
(ii) All incidental noise in the hall is focused at speaker.

Echoes:

It is found that if the interval between the direct sound and reflected sound is less than 1/15 sec, the reflected sound is helpful in increasing the loudness but those sounds arriving later cause confusions and should be weakened as far as possible. In big halls with high ceiling, the surfaces which may give echo should be covered with sound absorbing material to prevent reflection from them.

 Zones of maximum and minimum sound:

In a hall there may be zones of maximum and minimum sound. This is due to curved reflecting surfaces of a hall which produce undesirable concentration of sound in some regions of hall and reduce intensity at other places. In an acoustically good hall, this is undesirable. Therefore, the curved surfaces should be avoided or covered with sound absorbing materials.

Proper reverberation:

It is common experience that sound of a speaker persists unduly in the hall. The sound reaches the listener directly and then after successive reflections from the walls, ceiling etc. The listener therefore receives a series of sounds of diminishing intensity. The persistence of sound is called reverberation.

Professor Sabine measured intensity of sound in a hall with time. If a graph is plotted between intensity and time, it is as shown in fig. When sound is started, it grows to a maximum value and when it is stopped it decays to zero in exponential manner: (i.e. first rapidly and then slowly). Clearly if a syllable persists for a time during which the next is uttered, it will overlap. The most important parameter to be controlled is reverberation time which is roughly the time during which the sound persists in a hall. More precisely reverberation time is nearly the time during which the sound intensity; falls to the minimum audible Value (called audibility threshold) from an initial value which is 106 time the audibility threshold. Thus, the reverberation time
is the time taken for the sound to fall 10-6 time of its maximum steady value. If the period of reverberation is large, then at following sequence of short sounds such as syllables in speech will ova lap. The greater is the time of reverberation of a half the greater will be the confusion due to mixing different syllables. As a matter of fact excessive reverberation is the common defect in acoustics large buildings. By decrasing time of reverberation, the loudness of the hall also decreases.
If the reverberation is very small, the loudness is insufficient. In such a hall the speaker finds no response from the hall. Such a hall is said to be dead for the speaker. In such a hall the sound falls below audibility very soon.
Thus reverberation period should neither be too large nor small, but it must possess a suitable value which may be good for speaker and the audience. This suitable value of time is called optimum reverberation time for the hall. Sabine showed that the optimum period of reverberation depends on the size of the hall and the quality of sound.
After measuring reverberation time (T) in various rooms halls of different sizes, Sabine gave an empirical relation.

T = \dfrac{CV}{A}

where C is a constant, V is volume of hall in m3 and A is total absorption of sound in a hall.

A = \sum a s
where a’s are absorption coefficients of surfaces of surface areas of different materials present in the hall. Clearly reverberation time can be reduced if the total absorbing area A is increased.
That is why the reverberation period of an will be more than that of a crowded hall.
For a given hall; the reverberation time can be controlled by inserting or removing sound absorbing materials at the walls.
The period is greater for music than for speech. For example, for a hall of 104 m3, the reverberation period is between 1.0 and 1.5 sec for speech and between 1.5 and 2.0 sec for music. Below this value the intensity of sound is weak and above this value, the syllables overlap.

 Insulation of hall from noise:

The building should be insulated properly from external and internal noises. To eliminate external noise the doors and windows should be tightly fitted. Double doors and windows with separate frames for each having insulating material between them should be used.
The inside noise of typewriters, machinery etc. is reduced by placing them on absorbing materials like pads, wood, felt etc. and using curtains of absorbing materials.

]]> http://oscience.info/physics/dopplers-effect/dopplers-effect/feed/ 0 Principle of superposition http://oscience.info/physics/principle-of-superposition/ http://oscience.info/physics/principle-of-superposition/#comments Wed, 04 Apr 2012 19:07:01 +0000 sandeep http://oscience.info/?p=6677 Principle of superposition

When two or more waves propagating in a medium arrive at the same point simultaneously, a new wave is produced. This phenomenon is called superposition of According to Young the net displacement at any point of the medium is equal to the algebraic sum of displacements of individual waves arriving at that point simultaneously. This is called the principle of superposition and holds good as long as the amplitude of the waves is not too large. This principle is of extreme importance and can be applied to many types of waves e.g. sound waves, light waves, wave pulses etc. The superposition of harmonic waves gives rise to interference, beats and standing waves.
To recognize the types of waves, the following
equations must be known.

Equation of a straight line

Equation of straight line

y = mx + c

Equation of circle of radius ‘a’ is x2 + y2 = a2

Equation of circle

Equation of ellipse

 

Equation of ellipse


\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1

Where ,

a = semi-major axis

b = semi-minor axis

Equation of parabola is

y2 = 4 a x ( fig .a )

Equation of parabola

This is parabola symmetrically about x-axis and

x2 = 4 a y (fig . b)

This is parabola symmetrical about y – axis

Beats:

When two waves of slightly different frequencies travel along the same straight line and along the same direction, they superimpose in such a way that the resultant intensity alternatively increases or decreases. This phenomenon of waxing and waning of sound is called beats. One waxing and one waning forms one beat.
For the sake of simplicity we assume the two waves of slightly different frequencies n1 and n2 (at x =0 ) are represented as

y_1 = a sin 2 \pi n_1 t \text{and} \, y_2 = a \sin 2 \pi n_2 t

From Young’s principle of superposition

y = y_1 + y_2 = a \sin 2 \pi n_1 t + a \sin 2 \pi n_2 t \\ 2a \sin \dfrac{2 \pi ( n_1 + n_2 ) t}{2} \cos \dfrac{2 \pi ( n_1 - n_2 ) t}{2} …..(1)

Let n1 = n and n2 = n + \triangle n such that

\triangle n << n ; then equation (1) gives

y = 2a \sin 2 \pi n t \cos \pi (n_1 - n_2 )t \\ \text{or} \, \, \, y = 2a \cos \pi ( n_1 - n_2 ) t \sin 2 \pi n t \dots (2a) \\ \, = A \sin 2 \pi n t \\ \text{where} A = 2a \cos \pi ( n_1 - n_2 )t is the amplitude of resultant and obviously depends on time.

The amplitude A = 2a \cos \pi ( n_1 - n_2 ) t ….(3)

is maximum for \cos \pi ( n_1 - n_2 ) t = \pm 1 \\ \text{or} \pi (n_1 - n_2 )t = r \pi where r is an integer.

The instant of maximum are given by

t = \dfrac{r}{n_1 - n_2} = 0 , \dfrac{1}{n_1 - n_2} , \dfrac{2}{n_1 - n_2} , \dfrac{3}{n_1 - n_2}, \dots (4)

Obviously time interval between two consecutive maxima

= \dfrac{1}{n_1 - n_2}

Frequency of maxima = ( n1 – n2 ) sec -1

The amplitude is minimum for \cos \pi ( n_1 - n_2 ) t = 0 \\ \text{or} \pi ( n_1 - n_2 ) t = ( 2r + 1 ) \dfrac{\pi}{2} , r = 0,1,2,3, \dots \\ \text{The instant of minima are given by } \\ t = ( r + \dfrac{1}{2} ) ( \dfrac{1}{n_1 - n_2 )} \\ = \dfrac{1}{2 ( n_1 - n_2 )} , \dfrac{3}{2 ( n_1 - n_2 )} , \dfrac{5}{2 ( n_1 - n_2 )} \dots

The time interval between two consecutive minima   = \dfrac{1}{n_1 - n_2} \\ \therefore \text{Frequency of minima} \, = ( n_1 - n_2) \text{sec} ^{-1}

The number of beats produced per second.

= n_1 \to n_2

Remark: When the prongs of a turning fork are loaded, the frequency of fork decreases and when they are filed; the frequency of fork increases.

Stationary Waves:

When two wave trains of same frequency and amplitude travel with the same velocity along the
same straight line in opposite directions, they superimpose to, produce a new type of wave called stationary wave or standing wave.

The name stationary for such type of waves is justified because there is no flow of energy along the
wave. Let the incident wave propagating along Y- axis be

y_1 = a \sin ( \omega t - kx ) …(1)

and the wave reflected from the boundary traveling along negative X-axis is

y_2 = \pm a \sin ( \omega t - kx )….(2)

The positive and negative signs are used: if the boundary is free or rigid respectively.

 Case (i). If boundary is free: then equation (2) is,

y_2 = a \sin ( \omega t + kx ). ….(3)

The resultant displacement due to these incident and reflected waves is

y = y_1 + y_2 \\ = a \sin ( \omega t - kx ) + a \sin ( \omega t + kx ) \\ = 2a \sin \dfrac{\omega t - kx + \omega t + kx}{2} \cos \dfrac{\omega t - kx - \omega t - kx }{2} \\ = 2a \sin \omega t \cos kx = 2a \cos kx sin \omega t \dots (4) \\ = A \sin \omega t ….(5)

Where A = 2 a \cos kx = 2a \cos \dfrac{2 \pi x}{\lambda} \dots (6) \\ \, ( \text{since} k = \dfrac{\omega }{v} = \dfrac{2 \pi}{\lambda} )

is the amplitude of resultant wave.

At positions where \cos \dfrac{2 \pi x}{\lambda} = \pm 1 , the displacement is maximum. Such points are called antinodes and are given by

\dfrac{2 \pi x}{\lambda} = 0 , \pi , 2 \pi , 3 \pi , \dots = r \pi ( r = 0,1,2,\dots ) \\ \therefore \, \, \, \, \, x = \dfrac{r \lambda}{2} = 0 , \dfrac{\lambda}{2} , \lambda , \dfrac{3 \lambda}{2} , 2 \lambda \dots ( 7 )

The separation between two consecutive antinodes is \dfrac{\lambda}{2}
At positions where \cos \dfrac{2 \pi x}{\lambda} = 0, the displacement is always zero. Such points are called nodes and are given by

\dfrac{2 \pi x}{\lambda} = \dfrac{\pi}{2} , \dfrac{3 \pi}{2} , \dfrac{5 \pi}{2} = ( 2r + 1 ) \dfrac{\pi}{2} ( r = 0,1,2,\dots ) \\ \therefore \, \, x = ( 2 r + 1 ) \dfrac{\lambda}{4} = \dfrac{\lambda}{4} , \dfrac{3 \lambda}{4} , \dfrac{5 \lambda}{4} \, \, \, \, \dots (8)

The separation between two consecutive nodes is  \dfrac{\lambda}{2}
From (7) and (8) it is obvious that at free boundary always an antinodes is farmed. Midway between the antinodes, there are nodes.
The separation between a node and neighboring antinodes is \dfrac{\lambda}{2}

Case (ii)  If the boundary is rigid , then

y_2 = - a \sin ( \omega t + k x ) \, \, \, \, \, \dots ( 9 ) \\ \therefore\text{The resultant displacement} \\ y = y_1 + y_2 = a \sin ( \omega t - k x ) - a \sin ( \omega t + k x ) \\ = 2a \cos \dfrac{\omega t - k x + \omega t + k x}{2} \\ \, \, \, \, \, \, \, \sin ( \dfrac{\omega t - k x - \omega t - k x}{2} \\ = - 2a \sin kx \cos \omega t \, \, \, \, \, \dots (10) \\ = A \cos \omega t \, \, \, \, \, \, \dots (11) \therefore

Amplitude of resultant disturbance,

A = - 2 a \sin k x = - 2 a \sin \dfrac{2 \pi}{x} \, \, \, \dots (12)

The positions of maximum displacement or antinodes are

\sin \dfrac{2 \pi x}{\lambda} = \pm 1 \\ or \, \, \, \dfrac{2 \pi x}{\lambda} = ( 2r + 1 ) \dfrac{\pi}{2} ( r = 0 , 1 , 2 , 3 , \dots ) \\ \therefore \, \, \, x = ( 2r + 1 ) \dfrac{\lambda}{4} = \dfrac{\lambda}{4} , \dfrac{3 \lambda}{4} , \dfrac{5 \lambda}{4} \dots
The positions of zero displacement or nodes are

\sin \dfrac{2 \pi x}{\lambda} = 0 \text{or} \dfrac{2 \pi x}{\lambda} = r \pi ( r = 0 , 1 , 2 , 3 , \dots ) \\ \therefore \, \, \, x = \dfrac{r \lambda}{2} = 0 , \dfrac{\lambda}{2} , \dfrac{3 \lambda}{2} , \dfrac{5 \lambda}{2} \dots

Thus a node is always formed at rigid boundary lowest.

Fundamental tone, harmonics and overtones:

The sound of frequency produced by a musical instrument is called the fundamental tone. The sounds of other frequencies produced by the musical instrument are called overtones. The overtones whose frequencies are integral multiplies of the fundamental frequency are called the harmonics. The fundamental tone is also called the first harmonic. If first harmonic is n, then the tones of frequencies 2n, 3n, 4n … are called the second, the third and the fourth harmonic respectively. If frequencies of sound emitted by an instrument are n, 1.5n, 2n, 2.5n, 3n etc, then the notes of frequencies 1.5n, 2n, 2.5n, 3n are overtones, while those of frequencies 2n, 3n are second and third harmonics respectively.

Stationary Waves in Strings Fixed at Both Ends:

For transverse vibrations in string, we have

Speed of waves v = n \lambda = \sqrt{\dfrac{T}{M}}

where T is tension in string and m is mass per unit length of string

\therefore \text{Frequency} n = \dfrac{1}{\lambda} \sqrt{\dfrac{T}{m}} \, \, \, (1)

When the string is plucked in the middle, it vibrates in one loop with nodes at fixed ends and antinodes in the middle; so that length of string

tones

l = \dfrac{\lambda _1}{2} or \, \lambda _1 = 2l \\ \, \therefore \, \, \, n = n_1 = \dfrac{1}{\lambda} \sqrt{\dfrac{T}{M}} = \dfrac{1}{2l} \sqrt{\dfrac{T}{M}}

This tone is emitted is called the fundamental or first harmonic.

If the wire is plucked at one fourth of its length, the string vibrates in two loops, so that

l = \dfrac{\lambda _2}{2} + \dfrac{\lambda _2}{2} = \lambda _2 \\ \therefore \text{frequency} \\ n_2 = \dfrac{1}{\lambda _2} \sqrt{\dfrac{T}{m}} = 2 . \dfrac{1}{2l} \sqrt{\dfrac{T}{m}} = 2n

This tone is called the second harmonic or first overtone.
In the string is plucked at one sixth of its length, the string vibrates in three loops, so that

l = \dfrac{\lambda _3}{2} + \dfrac{\lambda _3}{2} + \dfrac{\lambda _3}{2} = \dfrac{3 \lambda _3}{2} \\ \therefore \text{frequency} \\ n_3 = \dfrac{1}{\lambda _3} \sqrt{\dfrac{T}{m}} = 3 . \dfrac{1}{2l} \sqrt{T}{m} = 3n

This tone is called third harmonic or second overtone.
In general when the string vibrates in p-loops, the frequency

n_p = \dfrac{1}{\lambda _p} \sqrt{\dfrac{T}{m}} = \dfrac{p}{2 l} \sqrt{\dfrac{T}{m}} = pn

This tone is called the pth. Thus in the case of string fixed at both ends, all harmonics even and odd are present.

Melde’s Law:

If N is frequency of tuning fork for a given length ‘l’ of a string vibrating in p-loops under 2; tension T, then Melde’s law states
T.p2 = constant or p \sqrt{T} = constant

Vibrations of Air columns in Organ Pipes

The minimum frequency produced in a pipe is called fundamental and other notes are called overtones.

Open organ pipe :

An antinodes is always formed at the open end. Accordingly different notes produced in open pipe are shown in fig. In fundamental mode if \lambda _1 is wavelength, then

l = \dfrac{\lambda _1}{2} or \, \, \, \, \lambda_1 = 2l \\ \therefore \text{fundamental frequency} \\ n = \dfrac{v}{\lambda_1} = \dfrac{v}{2l} \, \, \dots (1)

For first overtone if \lambda _2 is the wavelength , then

l = \dfrac{\lambda_2}{2} + \dfrac{\lambda_2}{2} = \lambda_2 \dots (2) \\ \text{frequency of first overtone} , \\ n_2 = \dfrac{v}{\lambda _2} = \dfrac{v}{l} = 2 \dfrac{v}{2l} = 2n \dots (3)

This frequency is double of fundamental frequency and is therefore called second harmonic.

For second overtone, if \lambda _3 is the wavelength, then

l = \dfrac{\lambda _3}{2} + \dfrac{\lambda _3}{2} + \dfrac{\lambda _3}{2} = \dfrac{3 \lambda _3}{2} \text{or} \lambda = \dfrac{2l}{3} \\ \therefore \text{frequency of second overtone} \\ n_3 = \dfrac{v}{\lambda _3} = 3 . \dfrac{v}{2l} = 3n \dots (4)

This is the third harmonic Frequencies

Frequencies n1 : n2 : n3 : … = 1 : 2: 3:…  i.e. in open organ pipe all harmonics even or odd are present.

Stationary Waves and Harmonics in Closed Organ Pipe:

A node is always formed at closed end and antinodes at open end. Accordingly different harmonies produced in closed organ pipe are shown in fig.

In fundamental mode, if\lambda _1 is the wavelength of vibrations, then

l = \dfrac{\lambda _1}{4} \text{or} \lambda_1 = 4l \\ \therefore \text{fundamental frequency} n_1 = \dfrac{v}{\lambda _1} = \dfrac{v}{4l} \dots (5)

 

Frequency of first overtone, if \lambda _2 is the wavelength , then

l = \dfrac{\lambda_2}{2} + \dfrac{\lambda _2}{4} = \dfrac{3 \lambda _2}{4} or \lambda _2 = \dfrac{4l}{3} \\ \therefore{frequency of first overtone} , \\ n_2 = \dfrac{v}{\lambda_2} = 3. \dfrac{v}{4l} = 3n

This frequency is three times of fundamental. Therefore in closed pipe the first overtone is third
harmonic.
For second overtone; if \lambda _3 is the wavelength, then

l = \dfrac{\lambda _3}{2} + \dfrac{\lambda _3}{2} + \dfrac{\lambda _3}{4} = \dfrac{5 \lambda _3}{4} \text{or} \lambda _3 = 4l \\ \text{frequency of second overtone ,} \\ n_2 = \dfrac{v}{\lambda _3} = 5 . \dfrac{v}{4l} = 5n

Thus in closed pipe

n1 : n2 : n3 : … = 1 : 3 : 5 : …

Hence in closed organ pipe only odd harmonics are present.

End Correction :
So far we have considered that the antinodes is formed exactly at the open end of the pipe; but actually due to finite momentum of the particles the reflection takes place a little above the open end; that is why the antinodes is formed a little above the open end. For this a correction is applied being known as end correction.This is denoted by ‘c’ and its value to 0.6r. r being radius of the pipe. If lo is the length of pipe, then for closed pipe ‘l’ is replaced by lo + e while for open pipe ‘I’ is replaced by 1o + 2e .

]]> http://oscience.info/physics/principle-of-superposition/feed/ 0 Wave motion http://oscience.info/physics/wave-motion/wave-motion/ http://oscience.info/physics/wave-motion/wave-motion/#comments Wed, 04 Apr 2012 16:54:03 +0000 sandeep http://oscience.info/?p=6675 Wave motion:

Wave:

Wave is defined as the continuous transfer of state from one part of the medium to the other with finite velocity without changing its form.
There are two types of waves :

(i) Mechanical waves:

The waves which require a material medium for their propagation are called mechanical waves. Sound waves in air, waves in stretched string are examples of mechanical waves.

(ii) Electromagnetic waves:

The waves which require no material medium for their propagation are called electromagnetic waves. The examples of electromagnetic waves are light waves, heat radiations, X—rays, \gamma-rays etc.

Characteristics of medium for Mechanical waves :

>The medium must be elastic.

>The medium must have inertia.

>The damping must be very small

That is why bell is made of metal and not of wood.

Types of Mechanical waves :

There are two distinct types of mechanical waves:

Transverse waves are the waves in which particles of the medium execute simple harmonic motion about their mean positions at right angles to the direction of propagation of wave. The example is a wave spreading over the surface of water when a stone is dropped in a pond. All electromagnetic waves (e.g. light waves, heat waves etc.) are transverse. For, the propagation of transverse waves in medium, the medium must be rigid. The waves within solids are transverse waves.

Longitudinal waves are the waves in which the particles of the medium execute simple harmonic motion about their mean positions along the direction of propagation of wave. The example is sound waves in air.
When a metallic rod is rubbed along the direction of length, the waves produced in the rod are longitudinal.

Progressive waves:

A progressive wave is one in which the disturbance in continuously transmitted along the direction of propagation of wave. These waves transmit energy from one place to another place.

Amplitude (A):

The amplitude is maximum displacement of the vibrating particle from its mean position.

Period (T): The period is the time taken by the vibrating particle in one vibration.

Wavelength (\lambda) :  The wavelength is defined as the distance traversed by the disturbance in time during which the vibrating particle makes one complete vibration. The relation between these quantities are

\lambda = v T , n = \dfrac{1}{T} , v = n \lambda

Speed of Sound Waves:

Sound is a form of energy which produces the sensation of hearing and propagates through mechanical waves.

Longitudinal waves :

The speed of Longitudinal waves is a medium depends on elasticity (E) and density (d) of given by

Newton’s formula v = \sqrt{( \dfrac{E}{d} ) } …..(1)

For longitudinal waves in a solid elasticity E is replaced by Young’s modulus, Y, so that

v = \sqrt{ (\dfrac{Y}{d} ) } ( = 5300 m/s for steel ) ….(2)

For longitudinal waves in a liquid  E is replaced by Bulk modulus K, so that

v = \sqrt{ (\dfrac{K}{d} ) } = 1450 m/s in water…..(3)

For longitudinal waves in a gas,
Newton assumed that the changes in air are isothermal
Isothermal elasticity ET = P

\therefore v = \sqrt{\dfrac{P}{d}}

For at NTP, P = 105 N/m2 , d = 1.29 kg/m3

\therefore v = \sqrt{\dfrac{10^5}{1.29}} = 280 m/s

which is incorrect.

Then Laplace assumed that changes in air are adiabatic

Adiabatic elasticity ES = \gammaP

Where \gamma = \dfrac{C_P}{C_V} and P is pressure, so that

v = \sqrt{\dfrac{\gamma P}{d}} \\ = \sqrt{\dfrac{\gamma RT}{M}} = 332 m/s for air at 0o C) …(4)

where T is absolute temperature, M is molecular weight of gas and R = 8.3 joule/gm-mol K) is gas constant for 1 mole. Obliviously the velocity of sound is maximum in solids and minimum in gases. Clearly at a given temperature

v \propto \sqrt{\dfrac{1}{M}}

Effect of various factors on velocity of sound in gases :

(i) Effect of pressure :

The velocity of sound is independent of pressure.

(ii) Effect of temperature:

The velocity of sound increases with increase of temperature

As v = \sqrt{\dfrac{\gamma RT}{M}} ; \text{therefore} v \propto \sqrt{T}

where T is absolute temperature.

If v0 is velocity of sound at 0°C, then the velocity of sound at in all gases is given by

\dfrac{v_t}{v_0} = \sqrt{\dfrac{t + 273}{273}} \\ \text{or} \, v_t = v_0 + \dfrac{v<sub>0</sub>}{546} t …(6)

For velocity of sound in air , v0 = 332 m/s

or,      vt = v0 + 0.61 t m/s

(iii) Effect of humidity:

The velocity of sound in air increases with increase of humidity since density of air decreases with increase of humidity and v \propto \dfrac{1}{\sqrt{d}}
the velocity of sound is maximum in hydrogen (= 1270 mls) being four times than that in oxygen.

(iv)Effect of nature of gas

As v \propto ( 1 / \sqrt{M} )the velocity of sound is maximum in hydrogen (= 1270 m/s) being four times than that in oxygen.

(v) Effect of velocity of wind:

The velocity of sound is greater in the direction of wind and smaller in an opposite direction.

(vi) Effect of frequency:

The velocity of sound in air is independent of the frequency of propagating wave.

The speed of transverse waves in a string is given by

v = \sqrt{\dfrac{T}{M}} = \sqrt{\dfrac{T}{x r^2 d}} \\ = \sqrt{\dfrac{T/xr^2}{d}} = \sqrt{\dfrac{\text{stress}}{d}}

Where

T = tension in string,
m = mass per unit length of string,                                                                                                                                                      r  = radius of string,                                                                                                                                                         d = density of string

Equation of General Harmonic Wave:

The equation of a progressive wave is expressed in the following equivalent forms:

y = A \sin \omega ( t - \dfrac{x}{v} ) \\ = A \sin (\omega t - kx ) \\ = A \sin 2 \pi ( \dfrac{t}{T} - \dfrac{x}{\lambda} \\ \text{where} k = \dfrac{\omega}{v} = \dfrac{2 \pi}{\lambda}

is propagation constant and points along positive X- axis. Let if the wave propagates along negative axis, x is replaced by so that above equations take the forms

y = A \sin \omega ( t + \dfrac{x}{v} ) \\ = A \sin (\omega t + k x ) \\ = A \sin 2 \pi ( \dfrac{t}{T} + \dfrac{x}{\lambda} )

In above equation there may be phase factor \phi so that the general equation of wave propagating along positive takes the form

y = A \sin [ \omega ( t - \dfrac{x}{v} ) + \phi ] \\ = A \sin [ ( \omega t - kx ) + \phi ] \\ = A \sin [ 2 \pi ( \dfrac{t}{T} - \dfrac{x}{\lambda} ) + \phi ]

The quantity in bracket {\phi} is called the phase of the wave.

A transverse wave propagates by means of crests and troughs and there is no change in pressure in a transverse wave. A longitudinal wave propagates by means of compressions and rarefactions and there is always a change of pressure along a longitudinal wave, given by

Excess pressure p = – E \dfrac{\triangle y}{\triangle x} = E \dfrac{u}{v}

Pressure amplitude, P_{\text{max}} = E \dfrac{U_{\text{max}}}{v} = \dfrac{EA \omega}{v}
where u is particle-velocity and v wave-velocity.
The pressure being maximum at compressions and minimum at rarefactions.

Intensity of a wave I \propto A^2 \propto P_{\text{max}}2

 Differential equation of Harmonic wave motion is \dfrac{d^2 y}{d x^2} = \dfrac{1}{v^2} . \dfrac{d^2 y}{dt^2}

v being velocity of wave.

Thus a harmonic wave, is of the form y = f ( x – vt ) or y = f ( x + vt ) and for this \dfrac{d^2 y}{d x^2} \text{and} \dfrac{d^2 y}{d t^2} must exit and satisfy the differential equation

\dfrac{d^2 y}{d x^2} = \dfrac{1}{v^2} \dfrac{d^2 y}{d t^2}

The equation of propagation of a wave pulse is

y = \dfrac{a}{b + ( x \pm vt )^2}

where a and b are constant (-)  sign is for pulse propagating along (+) X-axis and positive sign for a pulse propagating along negative X-axis, v is velocity of pulse propagation.

Relation between path difference \triangle and phase difference \delta is

\triangle = \dfrac{\lambda}{2 \pi} \delta

 Intensity of a wave is defined as the energy passing per second through unit area perpendicular to direction of propagation of wave and is given by I = 2 \pi ^2 n^2 A^2 d.v \text{Obviously} \, I \propto A^2 \text{and} I \propto n^2 \text{pressure amplitude} \propto A \text{and intensity} I \propto A^2 \therefore I \propto P_{max}2

Limits of audibility of sound

The limit of audibility of sound is from 20 Hertz to 20,000 Hertz. The sound haying frequency less than 20 Hertz is called infra-sonic and that having frequency greater than 20,000 Hertz is called ultrasonic. The ultrasonic can be heard by dogs, owls etc. The ultrasonic are produced by

(i) piezoelectric effect

(ii) magnetostriction method and

(iii) Galton’s whistle.

Ultrasonic are used to determine depth of mines and sea, stimulate the plant growth, kill bacteria and smaller animals like rats, frogs and fishes.
Reflection of Sound: Echoes :

As sound propagates in the form of waves, it exhibits the phenomenon of reflection and refraction. When sound wave travelling in a medium strikes the surface separating the two media, a part of incident wave is reflected back into first medium obeying ordinary laws of reflection while the remainder is partly absorbed and partly refracted (or transmitted) into second medium.
When sound wave is reflected a rarer medium or free boundary, there is no but the nature of sound wave is changed i.e. on reflection the compression is reflected back as rarefaction and vice versa.
If incident wave is y = a sin (wt — kx), then the equation of reflected wave takes the form

y = a' \sin ( \omega t - kx )

where a’ is new amplitude of reflected wave.
As wave reverses direction, x has been replaced by (-x).

When sound wave is reflected from a denser medium or rigid boundary, the phase of wave is reversed but the nature does not change i.e. on reflection compression is reflected back as compression and rarefaction as rarefaction.
The equation of reflected wave in this case takes the form

y = a' \sin ( \omega t + k x + \pi ) = - a'\sin (\omega t + kx )

Echoes

The common experience of sound reflection is echoes heard in large halls and in the neighborhood of hills. An echo is simply the repetition of speaker’s own voice caused by reflection at a distant surface e.g. a cliff, a row of buildings or any other extended surface.
If t is the time interval between production of sound from source and its echo at the site of source, then the distance between source and reflector (s) is given by

2s = vt

where v is the velocity of sound.
The persistence of hearing is 0.1 sec, therefore in order that an echo of short sound (e.g. shot of clapping) may be heard distinctly, the echo must come 0.1 sec later than the sound.

Articulate sound:

In case of articulate sound, it has been found that one can hear or pronounce distinctly not more than 5 syllables per second. Therefore for monosyllabic sound the minimum time interval between sound and its echo is 1/5 sec, for disyllabic and trisyllabic sounds it is 2/5 and 3/5 sec respectively and so on.
Multiple echoes are produced when there are several reflecting surfaces. The speaking tubes,
stethoscope, whispering gallery and sounding boards are based upon the reflection of the sound.

]]> http://oscience.info/physics/wave-motion/wave-motion/feed/ 0 Alternating Current and Electrical Devices http://oscience.info/physics/alternating-current-and-electrical-devices/ http://oscience.info/physics/alternating-current-and-electrical-devices/#comments Sun, 01 Apr 2012 16:32:04 +0000 Subash http://oscience.info/?p=6543

Alternating Current

 

 

An alternating current is one which changes in magnitude and direction periodically and is abbreviated as a.c.

 

Alternating current

Alternating current


The source of alternating emf may be a dyamo or an electronic oscillator. The alternating emf E at any instant may be expressed as:

E = E_0 sin \omega t \cdots equation \, \, 1

 

Where \omega is angular frequency of alternating emf and E_o is the peak value or amplitude of alternating emf.

The frequency of alternating emf, f = \dfrac{ \omega}{2 \pi}

 

And time period of alternating emf,

T = \dfrac{1}{f} = \dfrac{2 \pi}{ \omega}

 

The alternating current in a circuit, fed by an alternating source of emf may be controlled by inductance L, resistance R and capacitance C. Due to presence of element L and C, the current is not necessarily in phase with the applied emf. Therefore alternating current is, in general, expressed as:

 

I = I_o \, sin ( \omega t + \phi ) \cdots Equation \, \, 1

 

Where \phi is the phase which may be positive, zero or negative depending on the values of reactive components L and C.

 

The average and rms value:

The average value of AC over full cycle is zero since there are equal positive and negative half cycles.

( I_{av} ) _{full \, cycle} = 0 \cdots equation \, \, 3

 

The average value of AC over half cycle is given by:

( I_{av} ) _{half \, cycle} = \dfrac{2 I_o}{\pi} \cdots Equation \, \, 4

 

The root mean square value of AC is:

I_{rms} = \sqrt{ [ ( I^2 ) _{av} ]} = \dfrac{I_o}{ \sqrt{2}} \cdots Equation \, \, 5

 

Similarly the average value of alternating voltages are given by:

( E_{av} ) _{full \, \, cycle} = 0 \cdots equation \, 6

 

( E_{av} ) _{half \, \, cycle} = \dfrac{2 E_0}{\pi} \cdots equation \, 7

 

E_{rms} = \dfrac{e_o}{ \sqrt{2}} \cdots Equation \, \, 8

 

The rms value of alternating current can also defined as the direct current which produces the same heating effect in a given resistor. Due to this reason the rms value of current is also known as effective or apparent value of current:

\therefore I_{effective} = I_{virtual} = \dfrac{I_o}{ \sqrt{2}}

 

Similarly the rms value of alternating voltage is called the effective or virtual value of alternating voltage.

E_{virtual} = E_{rms} = \dfrac{E}{ \sqrt{2}}

 

Alternating current shows heating effect only. The ac meters are based n heating effect and measure rms values.

 

 

Impedance and Reactance

 

Alternating current in a circuit may be controlled by resistance, inductance and capacitance, while the direct current may be controlled only by resistance.

Impedance (Z): In alternating current circuit, the ratio of emf applied and consequent current produced is called the impedance and is denoted by Z.

Z = \dfrac{E}{I}

 

Physically impedance of ac circuit is the hindrance offered by the circuit to the flow of ac thought it.

Reactance (X): The hindrance offered by inductance and capacitance to the flow of ac in an ac circuit is called reactance and is denoted by X. thus when there is no ohmic resistance in the circuit; the reactance is equal to impedance. The reactance due to inductance alone is called inductive reactance and is denoted by X_L while the reactance due to capacitance alone is called the capacitive reactance and is denoted by X_c .

 

 

Impedances and Phases of Ac circuits containing different elements

 

In an ac circuit the current and applied emfs are not necessarily in same phase. The applied emf (E) and current produed (I) may be expressed as:

E = E_o \, sin \omega t \cdots Equation \, 1 and

 

I = I_o sin ( \omega t + \varphi ) with I_o = \dfrac{E_o}{Z} \cdots equation \, 2

 

Where E_o \text{and} \, \, I_o are peak values of alternating emf and current.

(i) Circuit containing Pure Resistance: If a circuit, fed by an alternating emf E = E_o \, sin \omega t , contains pure resistance R, then current is given by:

I = \dfrac{E}{R} = \dfrac{E_o sin \omega t}{R} = I_o sin \omega t

 

Where, I_o = \dfrac{E_o}{R} \cdots Equation \, \, 3

 

Circuit consisting pure resistance

Circuit consisting pure resistance


Circuit consisting pure resistance

Circuit consisting pure resistance


 

Compare this with standard equation (2), we note that Impedance of circuit, Z = R and phase lead of current over emf, \phi = o ,

That is in a purely resistive a.c. circuit the current and voltage are in same phase and impedance of circuit is equal to the ohmic resistance.

 

(ii) Circuit Containing Pure Inductance: If a circuit, fed by an alternating emf E= E_o \, \, sin \, \, \omega t , contains pure inductance L, then current is given by:

I =I_o sin ( \omega t - \dfrac{ \pi}{2} ) \cdots Equation \, \, 4

Where, I_o = \dfrac{E-o}{ \omega L}

 

Circuit Containing Pure Inductance

Circuit Containing Pure Inductance


Comparing this with standard equation (2), we note that:

Z = \omega L \text{and} \, \, \phi = - \dfrac{ \pi}{2} \cdots Equation \, \, 5

That is in a purely inductive circuit the current lags behind the applied voltage by an angle \dfrac{ \pi}{2} and impedance of the circuit is \omega L and obviously that is inductive reactance,

Z-l = X_l = \omega L \cdots Equation \, \, 6

 

Circuit Containing Pure Inductance

Circuit Containing Pure Inductance


 

(iii) Circuit Containing Pure Capacitance: Let a circuit pure capacitance and the applied alternating e.m.f. be:

E = E_o sin \omega t

 

The current in circuit is given by:

I = I_o sin ( \omega t + \dfrac{ \pi}{2} ) \cdots equation \, \, 7

 

Where,

I_o = \dfrac{E_o}{1 / \omega C}

 

Circuit Containing Pure Capacitance

Circuit Containing Pure Capacitance


Comparing this with standard equation 2, we note that:

 

Z = \dfrac{1}{ \omega C} \text{and} \varphi = + \dfrac{ \pi}{2} \cdots Equation \, \, 8

That us in a purely capacitive circuit the current leads the applied emf by an angle \dfrac{ \pi}{2} and the impedance of the circuit is \dfrac{1}{ \omega C} and obviously this is capacitive reactance:

Z_C = X_C = \dfrac{1}{ \omega C}

 

Circuit Containing Pure Capacitance

Circuit Containing Pure Capacitance

 


 

Circuit Containing Resistance and Inductance in series

 

 

Let a circuit containing resistance R and inductance L in series be fed with an alternating emf E_o sin \omega t .

Let I be the current flowing in the circuit and V_R ( = IR ) the potential difference across resistance and V_l ( = \omega L I )the potential difference V_L across inductance leads the current I by an angle \dfrac{ \pi}{2} .

Circuit Containing Resistance and inductance

Circuit Containing Resistance and inductance


Circuit Containing Resistance and inductance

Circuit Containing Resistance and inductance


Therefore, resultant voltage is given by:

E = \sqrt{ ( VR^2 + VL^2 ) } = \sqrt{ ( RI ) ^2 + ( \omega L I ) ^2}

 

\therefore \dfrac{E}{I} = \sqrt{R^2 + ( \omega L ) ^2}

 

= \sqrt{ R^2 + XL ^2}

 

Therefore, Impedance of R – L circuit,

Z = \dfrac{E}{I} = \sqrt{R^2 + XL^2}

It is obvious that the current lags behind the emf by an angle \phi given by:

tan \phi = \dfrac{V_L}{V_R} = \dfrac{X_L I}{RI} = \dfrac{X_L}{R}

 

 

Circuit Containing Resistance and Capacitance in series

 

 

Let a circuit containing resistance R and capacitance C in series be fed with an alternating emf E = E_o sin \omega t . Let I be the current flowing in the circuit, VR the potential difference across resistance and Vc the potential difference across capacitance.

 

Circuit Containing Resistance and Capacitance in series

Circuit Containing Resistance and Capacitance in series


Circuit Containing Resistance and Capacitance in series

Circuit Containing Resistance and Capacitance in series


 

The potential difference V_R and current I are in same phase and potential difference lags behind the current I (and hence V_R ) by angle \dfrac{\pi}{2} .

The resultant emf is given by:

 

E = \sqrt{ ( VR^2 + VC^2 )} = \sqrt{ (RI) ^2 + (XCL) ^2}

 

\therefore , Z = \dfrac{E}{I} = \sqrt{ (R^2 + Xc^2)}

 

The current leads the applied emf by an angle \phi given by:

 

tan \phi = \dfrac{V_C}{V_R} = \dfrac{X_C I}{RI} = \dfrac{X_C}{R}

 

 

Circuit Containing Inductance and Capacitance in series

 

 

Let a circuit containing inductance L and capacitance C in series be fed with an alternating emf E_o sin \omega t . Let I be the current flowing in circuit, V_L the potential difference across inductance L and V_c the p.d. across capacitance C.

 

Circuit Containing Inductance and Capacitance in series

Circuit Containing Inductance and Capacitance in series


Circuit Containing Inductance and Capacitance in series

Circuit Containing Inductance and Capacitance in series


The potential difference V_c lags behind the current by angle \dfrac{ \pi}{2} and the potential difference V_L leads the current by angle \dfrac{ \pi}{2} .

\therefore Resultant \, \, \, applied \, \, \, emf , E = V_c - V_l

 

= X_C I - X-L I

 

\therefore Impedance \, \, of \, \, circuit \, Z = \dfrac{E}{I} = X_c - X_L = ( \dfrac{1}{ \omega C} - \omega L )

 

The leading of current over applied emf \phi = \dfrac{pi}{2} .

 

In case X_c = X_L , Z = 0 , then

 

\dfrac{1}{ \omega C} = \omega L

 

\therefore Frequency \, \, f = \dfrac{ \omega}{2 \pi} = \dfrac{1}{ 2 \pi \sqrt{LC}}

 

This frequency is called the resonant frequency.

 

 

Circuit containing Resistance, Inductance and Capacitance in series

 

 

Let a circuit containing a resistance R, inductance L and capacitance C in series be fed with an alternating emf E = E_o sin \omega t. Let I be the current flowing in circuit, V_R , V_L \, \, and \, \, V_C , and the respective potential differences across resistance R, inductance L and capacitance C. The p.d. V_c is in phase with current I. The p.d. Vc lags behind the current by angle \dfrac{ \pi}{2} . The p.d. V_L leads the current by angle \dfrac{ \pi}{2} .

Therefore ,

Resultant applied emf,

 

Circuit containing Resistance, Inductance and Capacitance in series

Circuit containing Resistance, Inductance and Capacitance in series


Circuit containing Resistance, Inductance and Capacitance in series

Circuit containing Resistance, Inductance and Capacitance in series


E = \sqrt{ (VR^2 + ( V_C - V_L ) ^2}

 

The phase lead of current over applied emf.

tan \phi = \dfrac{V_C - V_L}{V_R} = \dfrac{ X_C I - X_L I}{RI} = \dfrac{X_C - X_L}{R}

 

I.E.

\phi = tan ^{-1} ( \dfrac{X_C - X_L}{R} )

It is obvious that:

(i) If X_C > X_L , the value of \phi is negative i.e, current leads the applied emf.

 

(ii) If X_C < X_L the value of \phi is negative i.e, current lags behind the applied emf.

 

(iii) If X_C = X_L , the value of \phi is zero i.e, current and emf are in same phase. This is called the case of resonance and resonant frequency is given by the condition X_C = X_L .

 

f_r = \dfrac{ \omega r}{2 \pi} = \dfrac{1}{ 2 \pi \sqrt{LC}}

 

At resonance impedance is minimum: Z_{min} = R and current is maximum I_{max} = \dfrac{E}{Z_{min}} = \dfrac{E}{R}

 

Circuit containing Resistance, Inductance and Capacitance in series

Circuit containing Resistance, Inductance and Capacitance in series

 


Power in an AC circuit

 

The power is defined as the rate at which work is being done in the circuit. In ac circuit, the current and emf are not necessarily in the same phase, therefore we write:

E = E_o sin \omega t , I = I_0 sin ( \omega t + \phi )

 

The instantaneous power,

P = EI = E_o sin \omega t , I_o sin ( \omega t + \phi )

 

= \dfrac{1}{2} E_o I_0 2 sin \omega t sin ( \omega t + \phi )

 

Transformer

 

A transformer is a device for converting high voltage into low voltage and vice versa.

There are two types of transformer.

1. Step up transformer: It converts low voltage into high voltage.

2. Step down Transformer: It converts high voltage into low voltage.

Transformer

Transformer

 

The principle of a transformer is based on mutual induction and a transformer always works on AC. The input is applied across primary terminals and output across secondary terminals.

The ratio of number of turns in secondary and primary is called the turn ratio i.e,

\dfrac{n_s}{n_p} = turn \, \, ratio \, \, n

 

If E_p and E_s are alternating voltage, i_p and i_s the alternating currents across primary and secondary terminals respectively, we have:

 

\dfrac{E_s}{E_p} = \dfrac{i_p}{i_s} = \dfrac{n_s}{n_p} = n

 

Efficiency of transformer:

 

\eta = \dfrac{output \, \, power}{input \, \, power} = \dfrac{P_{out}}{p_{in}} = \dfrac{E_s i_s}{E_p i_p}

 

The step up transformer is used to transmit power at high voltage to reduce line loss appreciably.

 

 

Induction Coil

 

An induction coil is based on the phenomenon of mutual induction and is used to produce a large emf from a source of low emf. An emf of the order of 50,000 V (but feeble current) may be achieved from 12 V battery (but high current).

 

Induction coil

Induction Coil


It consists of a primary coil P_1 P_2 containing a few turns of thick insulated copper wire wound on a laminated soft iron core. A secondary coil is wound on the primary coil and contains a large number of turns of thin copper wire. The make and break arrangement is provided by screw and soft iron hammer arrangement DH. When the current is passed by the help of battery in the primary coil, the iron core is magnetized and attracts the hammer, so the contact between screw and hammer is broken. The current in primary thus stops and iron core is demagnetized and the hammer is released and the contact is established again. A capacitor is connected across the air gap to prevent undue sparking and save the surfaces from being damaged.

When current is established in primary coil the flux linked with secondary increases and so a large emf is induced in the secondary. When the current breaks, the resistance of circuit becomes infinite and so time constant \dfrac{L}{R} becomes infinitely small and so the rate of fall of current becomes very much and consequently a very large emf is induced across the secondary. Thus two opposite emfs are induced, one at make and the other at break, but the induced emf at break is very much higher than that at make. The emf at make is insufficient to break the insulation of air gap across S_1 \, \, and \, \, S_2 , and discharge passes only at break. Hence current in the secondary is intermittent and unidirectional.

Induction coil is used in discharge tube to cause discharge of air/gas filled in it.

 

Generator or Dynamo

 

It is to device to convert mechanical energy into electrical energy. The generator is based on the principle of electromagnetic induction. According to which when a coil rotates in a uniform magnetic field, an alternating emf is induced in it.

An ac generator consists of:

(i) Armature: It is rectangular coil (ABCD) carrying a large number of turns and wound on a soft iron core. The soft iron core is used to increase the magnetic flux.

Armature

Armature


(ii) Field magnet: It is a strong magnet having two poles N and S. The armature is rotated between the poles so that the axis armature is perpendicular to the magnetic field lines.

(iii) Slip Rings C_1 and C_2 : The leads of armature-coil are connected to two rings C_1 \, \, and \, \, C2 called the slip rings. The slip rings also rotate with the coil.

 

(iv) Brushes: The two brushes B_1 \, \, and \, \, B_2 are made of graphite and they touch the slip rings C_1 \, \, and \, \, C_2 permanently. As the rings rotate, the brushes remain in constant touch with the rings, the brushes are connected to the two terminals, T_1 \, \, and \, \, T_2 . The external circuit is connected to these terminals. The emf induced in the coil e = NBA \omega sin \omega t = e_o sin \omega t where e-o = NBA \omega is the peak voltage.

 

Motor

 

A motor is a device which converts electrical energy into mechanical energy. The principle of a dc motor is based on interaction of current and magnetic field i.e. a current carrying coil placed in a uniform magnetic field experiences a torque.

A dc motor consists of a field magnet, an armature, the slip rings and brushes. The arrangement is same as that of a dynamo.

When current is passed in the armature coil through the brushes, the coil experiences a torque. This torque rotates the coil which is on the shaft to which the mechanical load is attached. .

When the coil rotates, the induced emf e is produced which opposes the applied emf E. That is why the induced emf is also called the back emf.

So net emf = E – e

If R is the resistance of circuit, then current at any instant I = \dfrac{E - e}{r}

The induced emf is proportional to angular speed ( \omega ) of coil. When motor is at full speed, the back emf is very high and a low current flows through the armature.

 

Choke

 

The Choke coil is a coil of high inductance and low resistance. It is used to control current in ac circuit without any appreciable power loss. The use of choke coil is preferred over resistance because the powerless in choke coil is negligible. The average power in ac is given by:

P = E_{rms} I_{rms} cos \phi

Where,

cos \phi = \dfrac{R}{Z}

In a choke coil R \rightarrow 0 . This implies that cos \phi = 0 or power loss in choke coil is nearly zero.

 

Starter

 

The armature-resistance of a motor is low to reduce energy losses. At the start of motor, the back emf (e) is zero, so current may be too heavy (I=E\R) and may break the insulation of coil. For this reason, a large variable resistance called the starter is introduced in series with the armature.

Therefore at the start, a resistance is included in the circuit and moderate current flows. As motor picks up speed, the variable resistance is gradually withdrawn so that moderate current flows. If the electric supply suddenly fails, a large current flows due to back emf which may damage the armature. Therefore the starter resistance is automatically introduce at ‘on’ and ‘off’ of a motor.

 

Skin effect

 

When a wire carries a direct current (dc), it is distributed uniformly throughout the whole cross-section of the wire. But if the wire carries alternating current of high frequency, it is merely concentrated on outer layers and if the frequency of ac is very high, the current is almost wholly confined to the surface layer of the wire.

This phenomenon is called the skin effect. Due to reduction of effective cross-sectional area a conductor offers high resistance to ac. Hence in the case of ac, current carrying conductor is in the form of strands of fine wire connected in parallel at their ends.

]]> http://oscience.info/physics/alternating-current-and-electrical-devices/feed/ 0 Nuclei http://oscience.info/physics/nuclei/ http://oscience.info/physics/nuclei/#comments Fri, 16 Mar 2012 16:13:28 +0000 Subash http://oscience.info/?p=6514 Composition of Nucleus

 

The atom consists of central nucleus, containing entire positive charge and almost entire mass. According to accepted model the nucleus is composed of protons and neutrons.

The proton was discovered by Rutherford by bombardment of \alpha -particles on nitrogen in accordance with the following equation :

 

Composition of Nucleus

Composition of Nucleus


The superscripts (on the right) denote the mass number and subscripts (on the left) denote the atomic number.

The neutron was discovered by Chadwick by the bombardment of \alpha -particles on beryllium in accordance with:

 

Composition of Nucleus

Composition of Nucleus


[A neutron is neutral (zero charge) and mass number is 1].

The number of protons in a nucleus is called atomic number while the number of nucleons (i.e., protons neutrons) is called the mass number (A). In general mass number > atomic number (except for hydrogen nucleus when A = Z).

Due to being neutral, neutron is used for artificial disintegration.

 

Atomic masses

 

The masses of atoms, nuclei etc are expressed in terms of atomic mass unit (amu) represented by amu or ‘u’. For this mass of C-12 is taken as standard.

1 u = \dfrac{mass \, \, of \, \, carbon - 12 \, \, atom}{12}

 

=1.660565 \times 10^{-27} kg

 

Mass \, \, of \, \, proton ( m_p ) = 1.007276 u

 

Mass of Neutron ( m_n ) = 1.008665 u

 

Mass of electron ( m_e ) = 0.000549 u

 

Isotopes, isobars and isotones

 

The nuclides having same atomic number (Z) but different mass number (A) are called isotopes. The nuclides having same mass number (A), but different atomic number (Z) are called isobars.

The nuclides having same number of neutrons are called isotones. A nuclide is represented as Z^X A ,being the symbol of element.

 

Size of Nucleus

 

The size of nucleus is of the order of 10^{-14} m . Most of nuclei are spherical in shape. According to experimental observations, the radius of nucleus of atom of atomic weight ‘A’ is given by:

R = R_0 A^{1 / 3} \, \, \, Where R_0 = 1.2 \times 10^{-15} = 1.2 Fermi

 

Mass Defect and Binding Energy

 

According to Einstein the mass and energy are equivalent i.e., mass can be converted into energy and vice versa. The mass energy equivalence relation is E = mc^2 .

Accordingly 1 Kg mass is equivalent to energy:

= 1 \times ( 3 \times 10^8 ) ^2

 

= 9 \times 10^{16} Joules

 

And 1 amu = \dfrac{1}{6.02 \times 10^{26}} Kg \, \, mass is equivalent to energy 931 MeV.

 

It is observed that the mass of a nucleus is always less than the mass of constituent nucleons (i.e., protons neutrons). This difference of mass is called the mass defect. Let M (Z, A) be the mass of nucleus, m_p = the mass proton and m_n mass of neutron, then the mass defect.

 

\Delta m = Mass \, \, of \, \, nucleons \, \, - Mass \, \, of \, \, nucleus

 

= Z m_p + ( A - Z ) m_n - M ( Z , A )

 

This mass defect is in the form of binding energy of nucleus, which is responsible for binding the nucleons into a small nucleus.

Therefore,

Binding energy of nucleus =( \Delta m ) c^2 ,

And binding energy per nucleon = \dfrac{ ( \Delta m ) c^2}{A}

 

 

Variation of binding energy per nucleon with mass number ‘A’

 

The graph in figure represents the average binding energy per nucleon in MeV against mass number A. It is observed that the binding energy for nucleon (except _2 He^4 , _6 C^{12} \, \, and \, \, _8 O^{16} ) rises first sharply, reaches a maximum value 8.6 MeV at A = 56 and then falls slowly, decreasing to 7.6 MeV for elements of higher mass number A = 240.

 

Variation of binding energy

Variation of binding energy


Nuclear Forces

 

The protons and neutrons inside the nucleus are held together by strong attractive forces. These attractive forces cannot be gravitational since forces of repulsion between protons >> attractive gravitational force between protons. These forces are short range attractive forces called nuclear forces. The nuclear forces are strongest in nature, short range and charge independent, therefore the force between proton-proton is same as the force between neutron-neutron or proton-neutron.
Yukawa tried to explain the existence of these forces, accordingly the proton and neutron do not have independent existence between nucleus. The proton and neutron are inter convertible through negative and positive \pi - mesons .

The existence of meson gives rise to meson field which gives rise to attractive nuclear forces.

The mass of \pi -meson = 273 X mass of electron.

 

Radioactivity

 

The phenomenon of spontaneous emission of radiation ( \alpha , \beta , \gamma etc ) by certain nuclei is called radioactivity.

Laws of Radioactive Disintegration:

 

(a) Rutherford – Soddy laws:

 

(i) Radioactivity is nuclear phenomenon. It is independent of all physical and chemical conditions.

 

(ii) The disintegration is random and spontaneous. It is a matter of chance for any atom to disintegrate first.

 

(iii) The radioactive substances emit \alpha \, \, or \, \, \beta -particles along with \gamma -rays. These rays originate from the nuclei of disintegrating atom and form fresh radioactive products.

 

(iv) The rate of decay of atoms is proportional to the number of un-decayed radioactive atoms present at any instant. If N is the number of un-decayed atoms in a radioactive substance at any time t, dN the number of atoms disintegrating in time dt, the rate of decay is \dfrac{dN}{dt} so that,

- \dfrac{dN}{dt} \propto N \, \, \, or \, \, \, \dfrac{dN}{dt} = - \lambda N \cdots Equation \, \, 1

 

Where \lambda is a constant of proportionality called the decayed disintegration constant,

Equation (1) results,

N = N_0 e^{- \lambda t} \cdots Equation \, \, 2

 

Where N_0 = initial number of un-decayed radioactive atoms.

 

(b) Displacement Laws:

 

(i) When a nuclide emits \alpha -particie, its mass number is reduced by four and atomic number by two,

I.e.

 

Displacement Law

Displacement Law


(ii) When a nuclide emits a \beta -particles, its mass number remains unchanged but atomic number increases by one.

Displacement Law

Displacement Law


Where \overline{v} is the antineutrino.

The \beta – particles is not present initially in the nucleus but is produced due to dis-integration of neutron into a proton,

I.e.

_0 n^1 \rightarrow _1 H^1 + _{-1} \beta ^0 + \overline{v} ( antineutrino )

 

When a proton is converted into a neutron, positive \beta – particles or positron is emitted.

_1 H^1 \rightarrow _0 n^1 + _1 \beta ^0 + v (neutrino )

 

(iii) When a nuclide emits a gamma photon, neither the atomic number nor the mass number changes.

 

 

Half-life and Mean life

 

 

The half-life period of a radioactive substance is defined as the time in which one-half of the radioactive substance is disintegrated. If N_0 is initial number of radioactive atoms present; then in a half-life time T, the number of un-decayed radioactive atoms will be N_0 / 2 and in next half N_0 / 4 and so on.

That is t = T ( half-life), N = \dfrac{N_0}{2}

\therefore From \, \, relation \, \, N = N_0 e^{- \lambda t} \cdots Equation \, \, 1

 

We get,

 

\dfrac{N_0}{2} = N_0 e^{- \lambda T}

 

Or \, \, \, e^{- \lambda T} = \dfrac{1}{2} \cdots Equation \, \, 2

 

From equation 1 and 2, we get,

\dfrac{N}{N_0} = e^{- \lambda t} = ( \dfrac{1}{2} ) ^{ \dfrac{t}{T}} \cdots Equation \, \, 3

 

Equation (3) is the basic equation for the solution of half-life problems of radioactive elements.

The half-life ‘T’ and disintegration constant \lambda are related as:

T = \dfrac{0.6931}{ \lambda} \cdots Equation \, \, 4

The mean life of a radioactive substance is equal to the sum of life times of all atoms divided by the number of all atoms,

I.e.

Mean life,

\tau = \dfrac{sum \, \, of \, \, life \, \, times \, \, of \, \, all \, \, atoms}{Total \, \, number \, \, of \, \, atoms} = \dfrac{1}{ \lambda} \cdots Equation \, \, 5

From equation 4 and 5, we get:

T = 0.6931 \tau \, \, I \, E \, \, T < \tau \cdots Equation \, \, 6

 

Activity of radioactive substance

 

The activity of a radioactive substance means the rate of decay (or the number of disintegrations / sec).

This is denoted by:

Activity of radioactive substance

Activity of radioactive substance


If A_0 is the activity at time t=0, then,

A_0 = \lambda N_0 \therefore \dfrac{A}{A_o} = \dfrac{N}{N_0} = e^{ - \lambda t}

I.e.

A = A_0 e^{- \lambda t} \cdots Equation \, \, 8

 

Units of radioactivity

 

(1) Curie: It is defined as the activity of radioactive substance which gives 3.7 X 10^{10}  disintegration/sec which is also equal to the radioactivity of 1 g of pure radium.

 

(2) Rutherford: It is defined as the activity of radioactive substance which gives rise to 10^6 disintegrations per second.

 

(3) Becquerell: In S.I. system the unit of radioactivity is Becquerell.

1 Becquerell = 1 disintegration/sec

 

Mass-Energy Relation

 

According to Einstein mass and energy are inter convertible through relation E = mc^2 .

This is called mass-energy equivalence relation. Accordingly 1 Kg \, \, mass = 9 \times 10{16} Joule and 1 u = 931 Mev.

Positron: It is antiparticle of electron. It has same mass but opposite charge as that of electron. It was discovered by Anderson.

Pair production: When a photon of high energy (greater than 1.02 MeV) approaches a heavy nucleus, its energy is converted into mass and a pair of particles electron and positron is produced. This phenomenon is called pair production.

Pair Annihilation: When an electron and positron come near each other, their whole mass is converted into energy in the form of two photons. These photons travel in opposite directions to conserve momentum. This phenomenon is called pair annihilation.

 

 

Nuclear Fission

 

The splitting of heavy nucleus into two or more fragments of comparable masses, with an enormous release or energy is called nuclear fission. For example when slow neutron are bombarded on _{92} U^{235} , the fission takes place according to reaction,

_{92} U^{235} + _0 n^1 \rightarrow _{56} Ba^{141} + _{36} Kr^{92} + 3 ( _o n^1 ) + 200 MeV

In nuclear fission the sum of masses before reaction is greater than the sum of masses after reaction, the difference in mass being released in the form of fission energy.

 

Remarks:

1. It may be pointed out that it is not necessary that in each fission of uranium, the two fragments are Ba^{141} \, \, and \, \, Kr^{92} are formed but they may be any stable isotopes of middle weight atoms. The most probable division is into two fragments containing about 40% and 60% of the original nucleus with the emission of 2 or 3 neutrons per fission.

2. The fission of U^{238} takes place by fast neutrons.

 

 

Chain reaction

 

If on the average more than one of the neutrons produced in each fission are capable of causing further fission, the number of fission taking place at successive stages goes on increasing at a rapid rate, giving rise to self-sustained sequence of fission known as chain reaction. The chain reaction takes place only if the size of the fissionable material is greater than a certain size called the critical size. There are two types of chain reactions.

 

(1) Uncontrolled chain reaction: In this process the number of fissions in a given interval on the average goes on increasing and the system will have the explosive tendency. This forms the principle of atom bomb.

 

Controlled chain reaction

 

In this process the number of fissions in a given interval is maintained constant by absorbing a desired number of neutrons. This forms the principle of nuclear reactor, consisting of the following parts:

(i) Fuel: This fuel is U^{235} \, \, or \, \, Pu^{239} \, \, or \, \, U_{233}

(ii) Moderator: A moderator is a suitable material to slow down neutrons produced in the fission. The best choices as moderators are heavy water ( D_2 0 ) and graphite (C).

(iii) Controller: To maintain the steady rate of fission, the neutron absorbing material known as controller is used. The control rods are made of cadmium or boron-steel.

(iv) Coolant: To remove the considerable amount of heat produced in the fission process, suitable cooling fluids, known as coolants are used. The usual coolants are water, carbon-dioxide, air etc.

(v) Reactor shield: The intense neutrons and gamma radiations produced in nuclear reactors are harmful for human body. To protect the workers from these radiations, the reactor core is surrounded by concrete, wall, called the reactor shield.

 

 

Nuclear Fusion

 

The phenomenon of combination of two or more light nuclei to form a heavy nucleus with release of enormous amount of energy is called the nuclear fusion. The sum of masses before fusion is greater than the sum of masses after fusion, the difference in mass appearing as fusion energy.

For example, the fusion of two deuterium nuclei into helium is expressed as

_1 H62 + _1 H^2 \rightarrow _2 H^4 + 21.6 MeV

Thus fusion process occurs at extremely high temperature and high pressure just at sun where temperature is 10^7 K .

 

Remarks:

1. For the fusion to take place, the component nuclei must be within a distance of 10^{-14} m m. For this they must be imparted high energies to overcome the repulsive force between nuclei. This is possible when temperature is enormously high.

2. The principle of hydrogen bomb is also based in nuclear fusion.

3. The source of energy of sun and other star’s is nuclear fusion.

 

There are two possible cycles:

(1) Proton – proton cycle:

 

Proton – proton cycle

Proton – proton cycle


 

(2) Carbon cycle:

Carbon cycle

Carbon cycle


Carbon cycles

Carbon cycles


The proton-proton cycle occurs at relatively lower temperature as compared to carbon cycle which has greater efficiency at higher temperature.

At the sun whose interior temperature is about 2 \times 10^6 K , the proton-proton cycle has more chances for occurrence.

 

 

Nuclear Holocaust

 

The estimate of aftereffect of an atomic or nuclear explosion is called the nuclear holocaust. If a fusion bomb (causing the fusion of isotopes of hydrogen, deuterium and tritium) explodes; then the nuclear holocaust will not only destroy every form life on earth but will also make this planet (earth) unfit for life for all times. The radioactive waste will hang like a cloud in earth’s atmosphere and will absorb sun’s radiations, thus causing a long nuclear winter.

On August 6, 1945 USA dropped an atom bomb on Hiroshima (Japan) which produced an explosion equivalent to 20,000 tons of TNT and the entire population of that place was either killed or seriously affected.

]]> http://oscience.info/physics/nuclei/feed/ 0 Reflection, Refraction and Total Internal Reflection http://oscience.info/physics/reflection-refraction-and-total-internal-reflection/ http://oscience.info/physics/reflection-refraction-and-total-internal-reflection/#comments Thu, 15 Mar 2012 18:08:26 +0000 Subash http://oscience.info/?p=6498 Ray optics as a limiting case of wave optics

 

In my optics (also called geometrical optics) the light is supposed to be propagating in straight lines, called the light rays, which are supposed to be formed of corpuscles. In wave optics light is supposed to be propagating in the form of waves.

The phenomena of interference, diffraction and polarization could only be explained by wave theory. The diffraction of light is the bending of light round the edges of the obstacle, due to which sharp images of the objects may not be seen.

The phenomenon of diffraction is more noticeable if the size of the object is comparable to the wavelength of light. According to wave theory the path of light may only be rectilinear approximately and not exactly. When the size of aperture becomes much greater than the wavelength of light, the light follows the straight line path. Therefore it may be concluded that the ray optics is a limiting case of wave optics.

 

Reflection

 

Laws of Reflection

The regular reflection follows the two laws:

1. The incident ray, the reflected ray and normal to surface at the point of incidence all lie in the same plane.

2. The angle of incidence (i) is equal to the angle of reflection (r’).

 

(i) Formation of Image by the plane mirror: The formation of image of a point object O by a plane mirror is represented in figure below.

The image formed ‘I’ has the following characteristics.

 

Formation of Image by the plane mirror

Formation of Image by the plane mirror


(a) The size of image is equal to the size of object.

(b) The separation of image from mirror formed behind the mirror is equal to the separation of object from the mirror i.e. OM = MI.

(c) The image is virtual, erect and laterally reversed.

 

Number of images in inclined mirrors:

 

Let \theta be the angle between two plane minors:

(i) If the object is placed asymmetrically between mirrors, no. of images n = 360 / \theta

(ii) If the object is placed symmetrically between mirrors and the value of 360 / \theta is even , then n = ( 360 / \theta ) -1 .

(1 is subtracted because two images coincide)

 

Refraction

 

When a ray of light falls on the boundary separating the two media, there is a change in direction of ray. This phenomenon is called refraction.

Laws of Refraction:

(i)The incident ray, the refracted ray and normal to the surface separating the two media all lie in the same plane.

(ii) Snell’s Law: For two media, the ratio of sine of angle of incidence to the sine of the angle of refraction is constant for a beam of particular wavelength I.e.

\dfrac{sin \, i}{sin \, r}= constant = \dfrac{ \mu _2}{ \mu _1} = 1 \mu _2

 

Where \mu_1 and \mu_2 are absolute refractive indices of ‘I’ and ‘II’ media respectively and 1 \mu_2 is the refractive index of second medium with respect to ‘I‘ medium.

 

Snell’s Law

Snell’s Law


As light flows reversible path, we have:

\dfrac{sin \, r}{sin \, i} = 2 \mu_1 \cdots Equation \, \, 1

 

Multiplying equation 1 and 2 we get:

2 \mu_1 \times 1 \mu_2 \, \, \, \, or \, \, 1 \mu _1 = \dfrac{1}{1 \mu_2} \cdots Equation \, \, 3

 

Also the frequency of light remains unchanged when passing from one medium to the other.

 

The refractive index of a medium is defined as the ratio of speed of light in vacuum to the speed of light in medium.

I.e.

\mu = \dfrac{Speed \, \, of \, \, light \, \, in \, \, vaccum}{Speed \, \, of \, \, light \, \, in \, \, Medium} = \dfrac{c}{v}

 

= \dfrac{v \lambda_{air}}{v \lambda_{medium}} = \dfrac{\lambda_{air}}{\lambda_{medium}} \cdots Equation \, \, 4

 

\lambda_{air} \, \, and \, \, \lambda_{medium} being wavelength of light in air and medium respectively.

 

\therefore \dfrac{sin \, i}{sin \, r} = \dfrac{\mu_2}{\mu_1} = \dfrac{c/V_2}{c/v_1} = \dfrac{v_1}{v_2} = \dfrac{\lambda_1}{\lambda_2} \cdots Equation \, \, 5

 

Formation of image by Refraction

 

According to Snell’s law if \mu_2 > \mu_1 , I > r. That is if a ray of light enters from rarer medium to a denser medium, it is deviated towards the normal and if \mu_2 < \mu_1 , i< r that is if the ray of light enters from denser to a rarer medium it is deviated away from the normal.

Accordingly if the ray of light starting from objects ‘O’ in denser medium travels along OP, it is deviated away from the normal along PQ. The ray PQ appears to come from ‘I’.

Thus ‘I’ is the virtual image of ‘O’. It can be shown that:

 

Formation of image by Refraction

Formation of image by Refraction


\mu = \dfrac{Real \, \, depth ( OM )}{Apparent \, \, depth ( M I )} = \dfrac{t}{t-x} \cdots Equation \, \, 6

 

Where ‘x’ is displacement or apparent shift.

\therefore The \, \, apparent \, \, shift, \, \, x = ( 1 - \dfrac{1}{\mu} t ) \cdots Equation \, \, 7

 

Refraction through a number of media

 

Now let us consider the refraction of light ray through a series of media as shown in figure. The ray AB is incident on air-water interface at an angle ‘I’. The ray is deviated in water along BC towards the normal.

Then it falls on water-glass interface and is again deviated towards normal along CD. If the last medium is again air, the ray emerges parallel to the incident ray. Let r_1 and r_2 be angles of refraction in water and glass respectively.

Then from Snell’s law,

 

Refraction through a number of media

Refraction through a number of media


\dfrac{sin i}{sin r} = \dfrac{\mu_w}{\mu_a} = _a \mu_w \cdots Equation \, \, i

 

\dfrac{sin \, r_1}{sin \, r_2} = \dfrac{\mu_g}{\mu_w} = _w \mu_g \cdots Equation \, \, ii

 

\dfrac{sin \, r_2}{sin \, i} = \dfrac{\mu_a}{\mu_g} = _g \mu_a \cdots Equation \, \, iii

 

Where,

\mu_a = refractive index of air = 1

\mu_w = refractive index of water

\mu_g = refractive index of glass

Multiplying equation (i), (ii) and (iii), we get:

_a \mu_w \times _g \mu_a = 1

 

_w \mu_g = \dfrac{1}{ _a \mu_w \times _g \mu_a} = \dfrac{_a \mu_g}{a \mu_w} \cdots Equation \, \, 8

 

Lateral shift on passing through a glass slab: Consider refractive of a ray AO incident on the slab at an angle of incidence ‘I’ through the glass slab EFGH.

After refraction the ray emerges parallel to the incident ray.

 

Refraction through a number of media

Refraction through a number of media


Let PQ be perpendicular dropped from P on incident ray produced.

The lateral displacement caused by plate,

X = PQ = OP sin (I – r)

\dfrac{OM}{cos r} sin (i-r) = \dfrac{t sin (i-r)}{cos r}

 

(iii) If ‘I’ is very small, r is also very small, then:

sin I \rightarrow I, \, \,sin r \rightarrow r \, \, and \, \, cos r \rightarrow 1

 

So that \dfrac{sin \, i}{sin \, r} = \mu takes the form \dfrac{i}{r} = \mu .

 

Therefore, the expression for lateral displacement takes the form:

x = \dfrac{t(i-r)}{1} = t \, I ( 1 - \dfrac{r}{i} )

 

= ( 1 - \dfrac{1}{\mu} t \, i )

 

Critical Angle: Total Internal Reflection

 

The angle of incidence in denser medium for which the angle of refraction in rarer medium is 90° is called the critical angle (C).

If \mu_r and \mu_d are refractive indices for rarer and denser media then,

\therefore \dfrac{sin \, i}{sin \, r} = \dfrac{\mu_2}{\mu_1} \, gives \\[3mm] \dfrac{sin \, C}{sin 90^o} = \dfrac{\mu _r}{\mu _d} = _d \mu_r

 

\therefore sin \, C = _d \mu_r = \dfrac{1}{_r \mu_d} = \dfrac{1}{\mu}

 

Where _r \mu_d = \mu it is the refractive index of denser medium with respect to rarer medium. When angle of incidence of the ray incident on rarer medium from denser medium is greater than the critical angle, the incident ray does not refract into rarer medium but is reflected back into denser medium. This phenomenon is called total internal reflection.

The conditions for total internal reflection are:

(i) The ray must travel from denser to rarer medium.

(ii) The angle of incidence i > critical angle C.

The critical angle for water-air, glass-air and diamond air interfaces is 49°, 42° and 24° respectively.

A fish or diver in water at depth h sees the whole outside world in horizontal circle of radius,

r = h tan C = \dfrac{h}{ \sqrt{ ( \mu ^2 - 1}}

 

\mu being refractive index of water.

 

Optical fibre

 

Optical fibre is a device based on total internal reflection by which signals may be transferred from one location to another. It is a thin pipe of plastic or specially coated glass in which light enters at one end and leaves at other end suffering a number of total internal reflections with little loss of energy.

 

Optical fiber

Optical fibre


The fibre works even if it is bent or twisted. For total internal reflection at the wall of fibre, the angle of incidence i > C, where sin C = 1 / _a n_f , _a n_f being refractive index of fibre with respect to air. The thickness of fibre is of the order of human hair =50 \mu = 50 \times 10^{-6} m .

A bundle of optical fibres can be put to several uses :

(1) It can be used as a ‘light pipe’ in medical and optical examination.

(2) It can transmit a laser or any other light beam.

(3) They are being used in telephone and other transmitting cables.

 

Sign Conventions

 

The sign conventions of coordinate geometry will be used, taking pole of mirror as origin. Accordingly the focal length of concave mirror is negative and that of convex mirror is positive.

The distance of object placed in front of mirror on the left (u) is (Negative X-axis) negative and the distance of image from mirror (v) is negative for real image and positive for virtual image.

 

Curved Mirrors and Mirror formulae

 

There are two types of spherical (curved) mirrors:

(i) Convex: Convex mirrors forms only virtual images of a real objects.

(ii) Concave: Concave mirrors may form real and virtual images or real objects.

Mirror formulae for all spherical mirrors are:

\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}

 

f = \dfrac{R}{2} and magnification \dfrac{I}{O} = - \dfrac{v}{u} = - \dfrac{f}{u-f}

For a convex mirror, f is positive and for a concave mirror f is negative.

If the object is to the left of the mirror u is negative and v is positive if image is on the right and negative if image is on the left of mirror.

 

Image formation by concave mirror

Image formation by concave mirror


Image formation by convex mirror

Image formation by convex mirror


It is observed that a spherical mirror of large aperture does not give a sharp image because the marginal rays (outer rays) are focused at a relatively smaller distance from pole P. This defect in image is called spherical aberration.

This is reduced by taking spherical mirror of small diameter as compared to its length focal or it is completely eliminated by taking parabolic mirrors.

 

Lens and Lens Formulae

 

There are two types of lenses:

(i) Convex (or converging) lens

(ii) Concave (or diverging) lens

 

Lens Maker’s Formula

 

If R_1 and R_3 are the radii of curvature of first and second refracting surfaces of a thin lens of focal length f, then lens-makers formula is

Lens Maker's Formula

Lens Maker's Formula


\dfrac{1}{f} = ( 1 \mu_2 -1 ) ( \dfrac{1}{R_1} - \dfrac{1}{R_1} )

 

= ( \mu - 1 ) ( \dfrac{1}{R_1 - \dfrac{1}{R_1} } )

 

Where 1 \mu_2 = \mu is refractive index of material of lens with respect to surrounding medium.

 

Thin lens formula is:

 

\dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u}

 

Magnification produced by a lens:

m = \dfrac{I}{O} = \dfrac{v}{u}

 

Where ‘I’ is size of image and O, is size of object.

If a lens (refractive index \mu_2 ) separates two media of refractive indices \mu_1 \, \, and \, \, \mu_3 then its total length ‘f’ is:

\dfrac{\mu_3}{f} = \dfrac{\mu_2 - \mu_1}{R_1} - \dfrac{\mu_3 - \mu_2}{R_2}

 

Thin lens formula

Thin lens formula


 

Power of lens:  The power, of a lens is its ability to deviate the rays towards axis and is given by:

P = \dfrac{1}{f ( in \, \, meters ) } Diopters = \dfrac{100}{f ( in \, \, cm ) } Diopeters

 

 

Lens immersed in a liquid: If a lens refractive index \mu_g is immersed in a liquid of refractive index\mu_l then its focal length ( f_I ) in liquid, is given by:

\dfrac{1}{f_1} = ( _1 \mu_g -1 ) ( \dfrac{1}{R_1} - \dfrac{1}{R_2} )

 

Where,

_1 \mu_g = \dfrac{\mu_g}{\mu_i}

 

(i) If f_ais the local length of lense in air, then,

f_l = \dfrac{n_g -1}{ \dfrac{n_g}{n_l} -1} \times f_a

 

Now there arise three cases:

(i) If \mu_g > \mu_l \, \, then f_l \, \, and \, \, f_a are of same sign but f_l > f_g .

That is the nature of lens remains unchanged, but its focal length increases and hence power of lens decrease. In other words the convergent lens becomes less convergent and divergent lens becomes less divergent.

(ii) id \mu_g = \mu_l , \, \, then \, \, f_l \rightarrow \infty . That us the lens behaves as a glass plate.

(iii) if \mu_g < \mu_l the f_l \, \, and \, \, f_q have opposite signs.

That is the nature of lens changes. A convergent lens becomes divergent and vice versa.

 

Newton’s formula

If the distances of object and image are not measured from optical center, but from first and second principal foci respectively, then,

Newton’s formula states f_1 f_2 = x_1 x_2

 

Newton’s formula

Newton’s formula


Where x_1 = F_1 O = distance of object from I principal focus F_1 .

x_2 = F_2 I = distance of image from II principal focus F_2 .

If medium on either side on lens is same, then f_2 = -f_1 = f

Therefore, newton’s formula takes the form, x_1 x_2 = -f^2 .

 

Lenses in contact: If two or more lenses of focal lenses of focal lengths f_1 , f_2 \cdots are placed in contact, then their equivalent focal length F is given by:

\dfrac{1}{F} = \dfrac{1}{f_1} + \dfrac{1}{f_2} + \cdots = \sum \dfrac{1}{f}

 

The power of communication P = p_1 + p_2 + \cdots = \sum P

 

Displacement method

 

For real image the distance between object and screen must be greater than or equal to 4f.

If the distance between object and screen (D) is greater than 4f, then there are two positions of the lens for which the image of object on the screen is distinct and clear. Using sign convention for real image formed by a lens, we have

\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u} \cdots Equation \, \, 1

Clearly u and v are interchangeable, i.e. in these two positions the distances of object and image from the lens are interchanged.

I.e.

v_1 = u_2 \, \, and v_2 = u_1

 

The figure above represents the formation of real images of an object in two positions P_1 \, \, and \, \, P_2 of lens L when D > 4f.

If I_1 \, \, and \, \, I_2 are the sizes of images in I and II positions of lens L, O is the size of object and m_1 , m_2 magnifications produced by lens in I position, and II position respectively, then,

m_1 = \dfrac{v_1}{u_1} = \dfrac{I_1}{O} and

 

m_2 = \dfrac{v_2}{u_2} = \dfrac{I_2}{O}

 

Multiplying (4) and (5), we get

m_1 m_2 = \dfrac{v_1}{u_1} \times \dfrac{v_2}{u_2} = \dfrac{I_1 I_2}{o^2}

 

But u_2 = v_1 = v \, \, and \, \, v_2 = u_1 =u

 

\therefore m_1 m_2 = \dfrac{v}{u} \times \dfrac{u}{v} = \dfrac{I_1 I_2}{O^2}

 

\therefore Size \, \, of \, \, object \, \, O = \sqrt{I_1 I_2} and focal length of lens.

 

\therefore f = \dfrac{D^2 - x^2}{4D} \cdots Equation \, \, 8

 

Displacement method

Displacement method


Silvering of one surface of lens

 

If one of the surfaces of a lens is silvered, the rays are first refracted by lens, then reflected from silvered surface and finally refracted by lens, so that the effective focal length of lens is:

\dfrac{1}{F} = \dfrac{1}{f_i} + \dfrac{1}{f_m} + \dfrac{1}{f_i} = \dfrac{2}{f_l} + \dfrac{1}{f_m}

 

Where f_l is focal length of lens and f_m is focal length of spherical mirror of radius of curvature of silvered surface.

This lens acts as a concave mirror and so the formula for image formation will be \dfrac{1}{F} = \dfrac{1}{v} + \dfrac{1}{u}

Two thin lenses separated by a distance: If two thin lenses of focal lengths f_1 , f_2 are placed at a distance d apart, then equivalent focal length of combination is:

\dfrac{1}{F} = \dfrac{1}{f_1} + \dfrac{1}{f_2} - \dfrac{d}{f_1 f_2}

 

Or, power of combination P = P_1 + P_2 - dp_1 p_2

]]> http://oscience.info/physics/reflection-refraction-and-total-internal-reflection/feed/ 0 Dispersion, Spectra and Optical instrument http://oscience.info/physics/dispersion-spectra-and-optical-instrument/ http://oscience.info/physics/dispersion-spectra-and-optical-instrument/#comments Thu, 15 Mar 2012 12:56:11 +0000 Subash http://oscience.info/?p=6484 Refraction through a Prism

 

A prism is a transparent medium enclosed by two plane refracting surfaces. Let EF be the monochromatic ray incident on the face PQ of prism PQR of refracting angle A at angle of incidence i_1 .

 

The ray is refracted along FG, r_1; being angle of refraction. The ray FG is incident on the face PR at angle of incidence r_2 and is refracted in air along GH. Thus GH is the emergent ray and i_2 is the angle of emergence. The angle between incident ray EF and emergent ray GH is called angle of deviation \delta .

Refraction through a Prism

Refraction through a Prism


 

Refraction through a Prism

Refraction through a Prism

 

For a prism if A is the refracting angle of prism then,

r_1 + r_2 = A \cdots equation \, \, 1 and

i_1 + i_2 = A + \delta \cdots equation \, \, 2

 

If \mu is the refractive index of material of prism then from Snell’s law:

\mu = \dfrac{sin i_1}{sin r_1} = \dfrac{sin i_2}{sin r_2} \cdots equation \, \, 3

 

If angle of incident is changed, the angle of deviation \delta changes as shown in figure. For a particular angle of incidence the deviation is minimum called angle of deviation \delta _m .

 

Minimum Deviation

 

At minimum deviation the refractive ray with in prism is parallel to the base of prism. So,

i_1 = i_2 = I ( say ) \\[3mm] r_1 = r_2 = r ( say )

 

Then from equation (1) and (2),

r + r = A

 

r = A / 2 \cdots equation \, \, \, 4a

 

I + I = A + \delta _m \\[3mm] I = \dfrac{A + \delta_m}{2} \cdots equation \, \, \, 4b

 

Therefore, the refractive index of material of prism:

\mu = \dfrac{sin i}{sin r} = \dfrac{sin \dfrac{A + \delta _m}{2}}{sin A / 2} \cdots equation \, \, 7

 

For a thin prism,

\delta _m = ( \mu - 1) A

 

Maximum deviation: For maximum deviation produced by a prism either i_1 \, \, \, or \, \, \, i_2 = 9060

 

Dispersion

 

The splitting of white light into constituent colors is called the dispersion. When white light falls on a prism, it is broken into constituent colors within the prism. So the emergent light has a number of colored beams, the violet being deviated most and red the least in visible region.

 

Dispersion

Dispersion


Thus the prism causes deviation as well as dispersion. If \delta _v \, \, \delta _r \, \, and \, \, delta_y are the deviation caused by prism is violet, red and mean yellow rays, then for small angled prism.

Angular \, \, dispersion \, \, = \delta _v - \delta _r = ( \mu _v - \mu _r ) A

 

Dispersive power,

\omega = \dfrac{Angular \, \, dispersion}{Mean \, \, deviation } = \dfrac{\delta _v - \delta _r}{\delta _y}

 

= \dfrac{ ( \mu _v - \mu _r ) A}{ ( \mu _y - 1 ) A} = \dfrac{ ( \mu _v - \mu _r}{\mu _y - 1}

 

Combination of two prisms

 

Two small prism may be combined to produce dispersion without deviation or deviation without dispersion.

(i) Dispersion without Deviation: In this arrangement of prism, this mean deviation ( \delta _y ) caused by one prism is cancelled by the mean deviation ( \delta' _y ) caused by the other prism.

I.e.

\delta _y + \delta' _y = 0 \, \, \, or \, \, \, ( \mu _y - 1 ) A + ( \mu'_y - 1 ) A' = 0

 

Or \, \, \, A' = - \dfrac{ \mu _y - 1}{\mu'_y - 1} A

 

Thus the angle of prism A’ in second prism has opposite sign as compared to angle A of first prism.

The net dispersion produced,

= ( \delta _v - \delta _r ) + ( \delta'_v - \delta'_r )

 

= ( \mu _v - \mu _r ) A + ( \mu'_v - \mu'_r ) A'

 

(ii) Deviation without Dispersion: In this arrangement of prism, the dispersion ( \delta _v - \delta _r ) caused by one prism is cancelled by dispersion ( \delta'_v - \delta'_r ) produced by the other prism.

I.e.

( \delta _v - \delta _r ) + |delta'_v - \delta'_r ) = 0

This gives,

 

A' = - \dfrac{\mu _v - \mu _r}{\mu'_v - \mu'_r}A

 

The net mean deviation = \delta _y + \delta'_y = ( \mu _y - 1 )A + ( \mu'_y - 1 ) A'

 

Spectra

 

The orderly array of colors (i.e. wavelengths) is called the spectrum. The spectra of bodies may be divided into two categories:

i. Emission spectrum: The spectrum of radiations emitted by a luminous body is called the spectrum. For example when light from a luminous electric tungsten bulb, live candle, luminous sodium vapor lamp is allowed to fall on a prism (or grating), the emission spectrum of that source is obtained. It is bright spectrum on dark back-ground.

ii. Absorption spectrum: When white light from a luminous source is first passed through an unexcited transparent substance (gas, liquid or solid) and then transmitted light is allowed to fall on the prism (or grating); the spectrum obtained is the absorption spectrum of that substance. The substance in unexcited state absorbs some radiations emitted from a luminous source. Hence in absorption spectrum certain wavelengths are missing; which appear as black in the spectrum. Hence the absorption spectrum contains dark part / lines.

The emission and absorption spectra may be divided into three subgroups:

 

Emission Spectrum:

 

(i) Continuous emission spectrum: It consists of continuous wavelengths (colors) in a definite wavelength range. It is obtained by incandescent solid or liquid in bulk state. It is independent of substance but depends on temperature only. The spectrum obtained from incandescent tungsten filament, live candle, burning coal, red hot metals is continuous.

(ii) Line emission spectrum: It consists of distinct bright lines and is produced by excited source in atomic state. For example the spectrum obtained from luminous helium, sodium, argon, mercury vapors is line emission spectrum.

(iii) Band emission spectrum: It consists of distinct bright band and is obtained by excited source in molecular state. For example the spectrum obtained by oxygen, nitrogen, carbon, cynogen etc is band emission spectrum.

 

Absorption Spectrum:

 

(i) Continuous absorption spectrum: It consists of absence of continuous wavelengths (or colors) in a definite wavelength range and is produced when the substance between the luminous body and the prism in unexcited bulk state. For example if we place red glass in between the luminous body and the prism, then all wavelengths are continuously absorbed except for red part of spectrum.

(ii) Line absorption spectrum : It consists of absence of distinct lines (i.e. dark lines) and is produced when the substance between the luminous body and the prism is in unexcited atomic state. For example if we place a sodium vapor lamp between tungsten filament and the prism, then two dark lines in yellow region of spectrum, appear. This is absorption line spectrum of sodium.

(iii) Band absorption spectrum: It consists of absence of distinct bands (i.e. dark bands) and is produced when the substance between the luminous body and the prism in unexcited molecular state. For example if we place dilute solution of potassium per magnate between luminous tungsten filament and the prism; two dark bands are observed in the spectrum.

 

Spectrometer

 

It is used to observe spectrum and to measure the deviation caused by the prism. A spectrometer essentially consists of three parts:

(i) Collimator

(ii) Prism table and

(iii) Telescope.

 

Spectrometer

Spectrometer


The collimator renders the rays parallel from an extended source which falls on prism and emerges as a parallel beam. The emergent parallel rays are received by the telescope.

 

Rainbow

 

Rainbow is an example of dispersion of sunlight by water drops in the atmosphere. The light suffers refractions and total internal reflections within the drops.

In the formation of primary rainbow: The light rays suffer two refractions and one total internal reflection. The innermost arc is violet and outermost is red.

Primary Rainbow

Primary Rainbow

 

Secondary rainbow

Secondary rainbow


 

In the formation of secondary rainbow: The light rays suffer two refractions and two total internal reflections. The innermost arc is red and outermost arc is violet. The secondary rainbow is broader than primary because light is weakened due to total reflections and angular dispersion is greater due to longer path within the drop. The secondary rainbow is situated above the primary because acute angles of deviations are greater than those for primary bow.

 

Scattering of Light

 

When light ray interacts with air molecules, its direction changes. This phenomenon is called scattering. If \lambda is the wavelength of light, then according to Lord Ray Leigh, the intensity of scattered light

I \propto \dfrac{1}{ \lambda ^4}

Accordingly blue light is scattered most and the red light is scattered least. Blue color of sky is due to scattering of light by air molecules. The red color of rising and setting sun is also due to scattering of light. At the time of rising and setting of sun, the red light traverses directly, while blue is scattered upward, hence rising and setting sun appears red.

 

Magnification

 

The size of an object depends on the angle subtended by object on eye. This angle is called visual angle. Greater is the visual angle; greater is the size of object. Stars are bigger than sun; but appear smaller because stars are much farther than sun and they subtend smaller angles on eye.

The angle subtended on eye may be increased by using telescopes and microscopes. The telescopes and microscopes form image of object. The image subtends larger angle on eye; hence the object appears big. The magnification produced by optical instrument (telescope/microscope) is defined as the ratio of angle ( \beta ) subtended by image on eye and the angle ( \alpha ) subtended by object on eye.

I.e.

Angular \, \, \, Magnification \, \, M = \dfrac{ \beta}{ \alpha}

 

Optical Instruments (Telescopes and Microscopes)

 

(i) Astronomical Telescope: It is used to see magnified images of distant objects. An astronomical telescope essentially consists of two co-axial convex lenses. The lens facing the object has large focal length and large aperture and is called objective, while the lens towards eye has small focal length aperture and is called eye lens.

[Capital letters symbolize for objective and small letters for eye lens i.e. F = focal length of objective, f = focal length of eye lens]

The magnifying power of telescope is:

 

= \dfrac{Angle \, \, subtended \, \, by \, \, final \, \, image \, \, at \, \, eye}{Angle \, \, subtended \, \, by \, \, object \, \, on \, \, eye} = \dfrac{ \beta}{ \alpha}

 

= ( m _0 \times m_e ) = - \dfrac{F}{f} ( 1 + \dfrac{f}{v} )

 

And length of telescope L = F + u

Where v = distance of final image from eye lenses

U =  Distance of real image A’B’ from eye lenses

Special cases (i) When final image is formed at a distance of distinct vision, then v = D

 

\therefore M = \dfrac{F}{f} ( 1 + \dfrac{f}{D} ) \, \, and \, \, L = F + u

 

(ii) When final image is formed at infinity, then v = \infty

 

Simple Microscope

 

It consists of a convex lens of small focal length f.

If \beta = angle subtended by image on eye

\alpha = angle subtended by object on eye, when object is at a distance of distinct vision (D)

 

Magnifying power,

M = \dfrac{ \beta}{ \alpha} = \dfrac{D}{v} ( 1 + \dfrac{v}{f} )

If final image is at \infty , v = \infty then M = \dfrac{D}{f} .

If final image is at distance of distance vision v = D, M = 1 + \dfrac{D}{f}

 

Compound Microscope

 

A compound microscope essentially consists of two co-axial convex lenses of small focal lengths. The lens facing the object is called objective while that toward eye is called eye lens.

Compound Microscope

Compound Microscope


Therefore, magnifying power of microscope,

M = \dfrac{ \beta}{ \alpha} ( m_0 \times m_e ) = \dfrac{v_0}{u_o} \dfrac{D}{v_e} ( 1 - \dfrac{v_e}{f_e} )

 

(0 symbolizes for objective and e for eye lens)

The length of microscope,

L = length of tube

= separation between lenses = v_0 + u_o

 

Special Cases

 

(i) When final image is formed at distance of distinct vision, v_e = D

\therefore M = - \dfrac{v_0}{u_o} ( 1 + \dfrac{D}{f_e} ) \, \, and \, \, L = v_0 + u_e

 

= \dfrac{L}{f_o} ( 1 + \dfrac{D}{f_e} )

 

(ii) When final image is formed at infinity, v_e = \infty , then

M_0 = - \dfrac{v_o}{u_o} \times \dfrac{D}{f_e} \, \, and \, \, L = v_0 + f_e

 

= - \dfrac{L}{f_o} \times \dfrac{D}{f_e}

 

 

Resolving power

 

The resolving power of an optical instrument is its ability to form distinct images of two neighboring objects. It is measured by the smallest angular separation between two neighboring objects whose images are just seen distinctly formed by the optical instrument. This smallest distance is called the limit of resolution.

Smaller is the limit of resolution, greater is the resolving power.

The angular limit of resolution of eye is 1′ or ( \dfrac{1}{60} ) ^0. It means that if two objects are so close that angle subtended by them on eye is less than 1’ or ( \dfrac{1}{60} ) ^0 , they will not be seen as separate.

Just Resolved

Just Resolved


Not Resolved

Not Resolved


The best criterion of limit of resolution was given by Lord Rayleigh. He thought that each object forms its diffraction pattern. For just resolution, the central maximum of one falls on the first minimum of the other. When the central maxima of two objects are closer than this objects appear over lapped and are not resolved but if the separation between then is more than this, they are said to be well resolved.

well Resolved

well Resolved


 

Telescope

 

If a is aperture of telescope and \lambda the wavelength, then resolving limit of telescope:

d \theta \propto \dfrac{ \lambda}{a}

For spherical aperture, d \theta = \dfrac{1.22 \lambda}{a}

Telescope

Telescope


Microscope: For microscope, \theta is the well resolved semi angle of cone of light rays entering the telescope, then limit of resolution \propto \dfrac{ \lambda}{ \theta} .

]]> http://oscience.info/physics/dispersion-spectra-and-optical-instrument/feed/ 0 Atoms and Molecules http://oscience.info/physics/atoms-and-molecules-2/ http://oscience.info/physics/atoms-and-molecules-2/#comments Thu, 15 Mar 2012 09:56:37 +0000 Subash http://oscience.info/?p=6476 Thomson Empirical Atom-Model

 

 

According to this model the mass and positive charge of an atom his uniformly distributed over the entire volume of atom and the electrons are embedded in this uniform distribution. This model is called the plumpudding model.

The model could explain the stability and neutrality of atom; but could not explain the discrete emission of radiations and the large angle scattering of \alpha -particles by the foil. As the model was empirical and had no experimental basis, therefore, it was rejected.

 

 

Rutherford Atom-Model

 

Rutherford performed the well-known experiment of scattering of \alpha-particles by thin gold foil and concluded :

(1) The atom is hollow.

(2) The entire positive charge and almost whole mass of the atom is concentrated in a small center called the nucleus.

(3) The electrons are too light to deflect the path of the \alpha -particles.

(4) The electrons revolve around the nucleus in circular orbits.

 

Rutherford scattering Formula

 

If N_{\emptyset}, is the number of \alpha -particles scattered at an angle \emptyset by a target of atomic number Z, then

N_{\emptyset} \propto \dfrac{Z^2}{sin ^4 \dfrac{\emptyset}{2}}

 

Rutherford scattering Formula

Rutherford scattering Formula


 

Size of Nucleus

 

If \alpha -particle of kinetic energy E_K hits a target nucleus directly, then its nearest distance of approach r_o is given by:

E_k = \dfrac{1}{4 \pi \epsilon _0} \dfrac{ ( Ze ) ( 2e ) }{r_o}

 

r_o = \dfrac{1}{4 \pi \epsilon _0} \dfrac{2 ze^2}{E_k}

 

This gives the size of nucleus. Its order is 10^{14} m.

Rutherford model could – not explain the stability of atom and the emission of discrete radiations. Since the model had experimental basis, therefore it was not rejected but modified by Bohr.

 

Bohr’s Atomic Model

 

The assumptions of Bohr’s model are:

(i) Circular orbits: The atom consists of central nucleus, containing the entire positive charge and almost all mass of the atom. The electrons rev the nucleus in certain definite orbits. The necessary centripetal force for circular orbit is provided by Coulomb’s attraction between the electron and nucleus,

I.e.

nv^2 = \dfrac{1}{4 \pi \epsilon _0} \dfrac{ (ze)(e)}{r^2} \cdots equation \, \, \, 1

 

Where m = mass of electron, r = radius of circular orbit, v = speed of electron in circular orbit, Ze = charge on nucleus, Z = atomic number, e= charge on electron= - 1.6 \times 10^{-19} \, \, Coulombs

 

(ii) Stationary orbits: The allowed definite orbits are those in which the electron does not radiate energy. These orbits are also called stationary orbits.

 

(iii) Quantum condition: The stationary orbits are those in which angular momentum of electron is intergral multiple of h/2 \pi ,

I.e.

mvr = n \dfrac{h}{2 \pi} \cdots n \, \, being \, \, integer \, \, \cdots equation \, \, \, 2

 

(iv) The nucleus is so heavy that its motion may be neglected.

 

(v) Constancy of mass: The mass of the electron in motion remains constant.

 

Deductions radii of orbits

 

Eliminating v from (1) and (2), we get the radius of nth stationary orbit, specified as T_n

r_n = \dfrac{\epsilon _0 h^2 n^2}{\pi m Ze^2} \cdots Equation \, \, 3

I.e.

r_n \propto n^2 that is radii of allowed orbits are proportional to the square of principal quantum number n. if we put the values of constants:

r_n = 0.53 \dfrac{n^2}{Z} \times 10^{-10} m = 0.53 \dfrac{n^2}{Z} A \cdots Equation \, \, 4

 

 

Velocity

 

If we eliminate r from (1) and (2), we get

v = \dfrac{Ze^2}{e \epsilon _0 hn} \propto \dfrac{Z}{n} \cdots Equation \, \, 5

 

This may be expressed as:

v_n = \dfrac{c}{137} \dfrac{Z}{n} \cdots equation \, \, 5a

 

 

Energy of electron

 

 

Kinetic energy,

E_k = \dfrac{1}{2}mv^2 = \dfrac{1}{4 \pi \epsilon _0} \dfrac{Ze^2}{2r} \cdots From \, \, Equation \, \, 1

 

The potential energy of electron,

U = \dfrac{1}{4 \pi \epsilon _0} \dfrac{ ( Z E ) ( - e ) }{r} = - \dfrac{1}{4 \epsilon _0} \dfrac{Ze^2}{r} \cdots Equation \, \, 6 a

 

Therefore,

Total energy, E = E_k + U

= - \dfrac{1}{4 \pi \epsilon _0} \dfrac{Ze ^2}{2r} = - E_k \cdots Equation \, 7

 

 

 

Energy Quantization

 

Total energy of electron in nth orbit,

E_n = - \dfrac{1}{4 \pi \epsilon _o} \dfrac{Ze^2}{2r}

 

= - \dfrac{1}{4 \pi \epsilon _0} \dfrac{Ze^2}{ ( \epsilon _0 h^2 n^2 / \pi m Ze^2 ) }

 

Or,

E_n = - \dfrac{Z^2 me^4}{\delta \epsilon ^2 _0 h^2 n^2}

 

= Z^2 ( \dfrac{me^4}{8 \epsilon _0 ^2 ch^3} ) hc = - \dfrac{Z^2 Rhc}{n^2} \cdots Equation \, \, 8

 

= - \dfrac{13.6 Z^2}{n^2} eV

 

WhereR = \dfrac{me^4}{8 \epsilon ^2 _0 ch^3} = 1.097 \times 10^7 m^{-7} is called Rydberg constant.

Obviously energy in nth orbit E_n \propto - \dfrac{1}{n^2}

I.e.

Energy is quantized.

 

Transitions

 

When as electron jumps from one another, the photon of frequency v is emitted or absorbed having energy equal to the difference of energies between initial and final states and being given by:

E_i - E_f = hv \, \, \, v = \dfrac{E_i - E_f}{h} \cdots Equation \, \, 3

 

 

Frequency of emitted radiation: if electron jumps from initial state n_i to a final state n_f then frequency of emitted or absorbed radiation v is given by:

E_i - E_f = hv

 

Or \, \, \, v = \dfrac{E_i - E_f}{h} = Z^2 R ( \dfrac{1}{n_f ^2} - \dfrac{1}{n^2_i} )

If c is the speed of the light and \gamma the wavelength of emitted or absorbed radiation, then;

v = \dfrac{c}{\gamma} = Z^2 Rc ( \dfrac{1}{n_f ^2} - \dfrac{1}{n^2 _i} )

This relation holds for radiations emitted by hydrogen like atoms. E.g. Singly ionized helium, doubly ionized lithium etc.

 

A note on Rydberg constant

 

Rydberg constant, according to Bohr’s theory is:

R = \dfrac{me^4}{8 \epsilon _o ^2 ch^3} = 1.097 \times 10^7 m^{-1}

 

In Bohr’s theory nucleus is assumed to be infinity heavy; but in actual practice it is not so. Therefore mass of nucleus must be taken into account. This is best accounted using the concept of reduced mass. If \mu is reduced mass of electron and nucleus, then:

\mu = \dfrac{m_em_n}{m_e + m_n}

 

Where m_n = mass of nucleus.

Then Rydberg constant will becomeR = \dfrac{ \mu e^4}{8 \epsilon _0 ^2 ch^3}

 

A \mu depends on mass of nucleus, which is different for nuclei of different elements. Hence Rydberg constant is different for different elements.

 

Hydrogen spectrum

 

Hydrogen spectrum is of two types:

(i) Emission spectrum

(ii) Absorption spectrum

The emission spectrum of hydrogen consists if 5 series.

(1)Lyman series: the lines of Lyman series are obtained when electron jumps from any higher states to ground (n = 1) state.

The wavelength of Lyman series are given by:

\dfrac{q}{\gamma} = R [ \dfrac{1}{I^2} - \dfrac{1}{n_i ^2} ]

 

Hydrogen spectrum

Hydrogen spectrum


For first member Lyman series n_i =2 , for second member n_i = 3 and so on and for limiting member n_i \propto \infty. The lines of Lyman series are found in ultraviolet region.

 

Balmer series

 

The lines of Balmer series are emitted when electron jumps from any higher state to n=2 state [i.e. from outer orbit to L-orbit].

The wavelengths of Balmer series are given by:

\dfrac{1}{\lambda} = R [ \dfrac{1}{2^2} - \dfrac{1}{n_i ^2} ] \, \, n_1 = 3 \, 4 \, 5 \, \cdots \, \infty

 

For first member n_i = 3 ,  for second member n_i = 4 and so on, and for limiting member n_i = \infty . The first, second, third, ……. Lines of Balmer series are called H_{\alpha} \, H_{\beta} \, H_{\gamma} ……. Lines respectively. The lines of Balmer series are found in visible region and this series was first discovered.

 

Pashen Series

 

The lines of Pashen series are emitted when electron jumps from any higher state to n = 3.

The wavelengths are given by:

\dfrac{1}{ \lambda} = R [ \dfrac{1}{3^2} - \dfrac{1}{n_i ^2} ] \, \, \, n_i = 4 \, \, 5 \, \, 6 \, \, \cdots \infty

For first member n_i = 4 , for second member n_i = 5 and so on and for limiting member n_i = \infty . The lines of Paschen series are found in infrared region.

 

 

Bracket Series

 

The lines of Bracket series are emitted when electron jumps from any higher state to n = 4.

The wavelengths are given by:

\dfrac{1}{\lambda} = R [ \dfrac{1}{4^2} - \dfrac{1}{n_i ^2} ] \, \, n_i = 5 \, \, 6 \, \, 7 \, \, \cdots \infty

 

This region is also found in infrared region.

 

 

Pfund series

 

This lines of Pfund series are emitted when electron jumps from any higher state to n = 5 state.

The wavelengths are given by:

\dfrac{1}{\lambda} = R [ \dfrac{1}{5^2} - \dfrac{1}{n_i ^2} ] \, \, \, n_i = 6 \, \, 7 \, \, \cdots \, \infty

This series lines in far-infrared region of spectrum.

Absorption spectrum: The absorption spectrum of hydrogen consist of only the Lyman series and its lines are obtained when electron jumps off from ground (n =1) state to any higher state.

 

 

Excitation and Ionization Potentials

 

The minimum energy required for electron-jump from ground state to any higher state is called excitation energy any the corresponding potential is called the excitation potential.

The first excitation energy = E_2 - E_1 and the first excitation potential= \dfrac{E_2 - E_1}{e} volt ,

Where e= electronic charge.

The second excitation energy = e_3 - E_1 the second excitation potential = \dfrac{E_3 - E_1}{e} volt.

The I th excitation energy =E_{I + 1} - E_1 and the second excitation potential v_i = \dfrac{E_i + 1 - E_1}{e}

The ionization energy =E_{ \infty} - E_1 and the ionization potential = \dfrac{E_{ \infty} - nE_1}{e}

 

Energy Levels of Hydrogen

 

The energy levels of hydrogen (Z = 1) are given by:

E_n = - \dfrac{13.6}{n^2} eV
Energy Levels of Hydrogen

Energy Levels of Hydrogen


Obviously the energy of hydrogen in ground orbit is least and goes on increasing with increase of order of orbit. The difference of energy between two successive levels also goes on decreasing.

The energy state of electron characterized by principal quantum number n=1 is called ground state and the corresponding orbit is called K- orbit. The energy state of electron characterized by n=2 is called first excited state and the corresponding orbit is called L-orbit. The energy state corresponding to n=3 is called second excited state and the orbit is called M-orbit and so on.

The energy state corresponding to n = \infty is called ionization state.

The energy of electron is quantized till it is in the orbit but if escapes from atom, it may have any energy. Therefore above n = \infty state, there is continuum; which represents that electron in this region may have any energy.

For hydrogen first excitation energy:

E_1 = E_2 - E_1 = - 3.4 - ( -13.6 ) = 10.2 eV

 

Second excitation energy E_{11} = E_3 - E_1

 

= [ - 1.51 - ( -13.6) ]

 

= 13.6 eV - 1.15 eV = 12.1 eV

 

E_{ion} = E_{\infty} - E_1 = 0 - ( - 13.6 ) = 13.6 eV

 

De-Broglie Model

 

According to the de-Broglie only those orbits are allowed as stationary orbits for electrons in which circumference of orbits is integral multiple of de-Broglie wavelength associated with electron.

I.e

2 \pi r = n \lambda

 

Where \lambda = \dfrac{h}{Pmv}

]]> http://oscience.info/physics/atoms-and-molecules-2/feed/ 0 Electrons and Photons http://oscience.info/physics/electrons-and-photons/ http://oscience.info/physics/electrons-and-photons/#comments Sun, 11 Mar 2012 12:30:35 +0000 Subash http://oscience.info/?p=6466 Cathode Ray

 

Cathode rays are the streams of electrons and were discovered by Sir William Crooke. They are produced by:

(i) A discharge tube containing gas at allow pressure of the order 10^{-3} mm of Hg. At this pressure the gas molecules ionize and the emitted electrons travel towards positive potential of anode. The positive ions hit the cathode to cause emission of electrons from cathode. These electrons also move towards anode. Thus the cathode rays in the discharge tube are the electrons produced due to ionization of gas and that emitted by cathode due to collision of positive ions.

(ii) An electron gun containing a filament fitted in a tube having a number of slits. The filament is heated by passing a current which may be controlled by a rheostat. The emitted electrons move towards slits under accelerated potential and emerge in the form of a collimated electron stream.

 

Properties of Cathode Rays:

 

1. Cathode rays are the streams of electrons. Therefore they carry negative charge.

2. The cathode rays are independent of the nature of the gas or electrodes employed in the discharge tube. Therefore e/m for cathode rays is universal constant:

= \dfrac{1.6 \times 10^{-19}}{9.1 \times 10^{-31}}

 

3. They can be deflected by electric and magnetic fields.

4. They have penetrating power and can penetrate through small thickness of matter (eg. Thin AL foil).

5. On striking the target of high atomic weight and high melting point, they produce X- rays.

6. They produce fluorescence on certain substance and hence affect photographic plate.

7. They have small ionizing power and ionize the gas through which they pass.

8. They travel in straight lines and cast shadow of objects placed in their path.

9. They produce heat when allowed to fall on matter.

10. They exert mechanical pressure so they can rotate a small paddle wheel.

11. They can exhibit interference and diffraction phenomena under suitable arrangements. Thus they may behave as waves.

 

Discovery of Electron

 

The electron was discovered by sir J.J. Thomson in 1897. He showed that cathode rays are simply the stream of electrons. Thomson was awarded Noble prize for this discovery. The mass of electrons is 9.1 \times 10^{-31} kg and the charge on electron is - 1.6 \times 10^{-19} coulomb.

 

Determination of e/m

 

The e/m of cathode rays was determined by Thomson by using crossed electric and magnetic fields.

If E and B are mutually perpendicular electric and magnetic fields and if an electron beam entering perpendicular to both the fields with velocity v remains undeflected, then:

\overrightarrow{f_e} + \overrightarrow{F_m} = 0

 

e v B = eE \, \, \, or \, \, \, v = \dfrac{E}{B} \cdots equation\, \, 1

 

‘E’ being charge of electron.

If ‘r’ is the radius of circular path of electrons in magnetic field only, then:

r = \dfrac{mv}{eE} \, \, or \, \, \dfrac{e}{m} = \dfrac{v}{rB} = \dfrac{E}{rB^2}

 

The value of \dfrac{e}{m} is called the specific charge and for electrons it comes out to be 1.76 \times 10^{11} coul \ kg .

 

 

Millikan’s Experiment

 

The charge on the electron was found by Millikan’s oil drop experiment.

Millikan’s Experiment

Millikan’s Experiment


In equilibrium of a charged oil drop between the region of plates:

qE = mg \rightarrow q = \dfrac{mg}{E}

As q = ne

Therefore,

\dfrac{mg}{nE}

 

Where electric field,

 

E = \dfrac{Potential \, \, difference \, \, between \, \, plates}{Seperation \, \, between \, \, plates}

 

= \dfrac{V}{d}

 

Electrical Conduction in Gases

 

The electrical conduction of gases is studied with the help of discharge tube. If pressure is reduced to zero atmospheric pressure, the following effects are observed.

(a) At atmospheric pressure no discharge takes place.

(b) Above 1 mm of hg: At a pressure more than 1 mm of Hg a luminosity is observed which is confined to each electrode, but major part of tube remains dark. This is called dark discharge.

(c) At 1 mm of Hg: When pressure is reduced to 1 mm oh Hg, along luminous column starting from anode fills the whole space. This is called positive column.

(d) At 0.5 mm of Hg: When the pressure is reduced to 0.5 mm of Hg, a colored glow is seen at cathode, called the cathode glow. Now the positive column breaks into a number of discrete patches of light, called striations. These are separated from each other by dark intervals. The region between negative column and striations remains dark; called Faraday’s dark space.

(e) At 0.1 mm of Hg : When the pressure is reduced to 0.1 mm of Hg, the striations positive column and negative glow move towards anode and so a dark space appears near the cathode, called Crooke’s dark space.

(f) At 10^{-2} - 10^{-3} mm of Hg : Finally when the pressure is reduced to 10^{-2} - 10^{-3} mm og Hg, the striations disappear and whole tube is filled with dark space. At pressure cathode rays are produced.

 

Electrical Conduction in Gases

Electrical Conduction in Gases


 

 

Quantum (Particle) Nature of Light

 

Some phenomena like photoelectric effect, Compton effect, Ramen effect could not be explained by Wave theory of light. Therefore quantum theory of light was proposed by Einstein who extended the Planck’s hypothesis to explain Black Body radiations. According to quantum theory of light  “light is propagated in bundles of small energy, each bundle being called a photon and possessing energy.”

\epsilon = h v = \dfrac{hc}{\gamma} \cdots Equation \, \, 1

Where ‘v’ is frequency, \gamma is wavelength of light and h is Planck’s constant = 6.62 \times 10^{-34} Joule-sec and c= speed of light in vaccum = 3 \times 10^8 m/s.

 

Momentum of photon,

p = \dfrac{hv}{c} = \dfrac{h}{\gamma} \cdots Equation \, \, 2

Dynamic or Kinetic mass of Photon,

m = \dfrac{hv}{Pc^2} = \dfrac{h}{c \gamma} \cdots Equation \, \, \, 4

The number of photons in a source of monochromatic radiation of wavelength \gamma and energy W or power P.

N = \dfrac{W}{\epsilon} = \dfrac{Pt}{\epsilon} \cdots Equation \, \, \, 5

 

Photoelectric Effect

 

The phenomenon of emission of electrons from a metallic surface by the use of light (or radiant) energy is called photoelectric effect. Caesium is best metal for photoelectric effect.

Characteristics of Photo-electric Effect:

 

(i) Effect of intensity: Intensity of light means the energy incident per unit area per second. For a given frequency, if intensity of incident light is increased, the photo-electric current increases and with decrease of intensity, the photo-electric current decreases, but the stopping potential remains the same.

Effect of intensity

Effect of intensity


Effect of intensity

Effect of intensity


This means that the intensity of incident light affect the photo-electric current but leaves the maximum kinetic energy of photo-electrons unchanged.

 

(ii) Effect of Frequency: When the intensity of incident light is kept fixed and frequency is increased, the photo-electric current remains the same; but the stopping potential increases.

If the frequency is decreased, the stopping potential decreases and at a particular frequency of incident light, the stopping potential becomes zero.

Effect of Frequency

Effect of Frequency


This value of frequency of incident light for which the stopping potential is zero is called threshold frequency v_o. If the frequency of incident (v) is less than the threshold frequency ( V_0 ), no photoelectric emission takes place.

Thus the increase of frequency increases maximum kinetic energy of photo-electrons but leaves the photo-electric current unchanged.

(iii) Effect of Photo metal: When frequency and intensity of incident light are kept fixed and photo-metal is changed, we observe that stopping potential ( V_s ) versus frequency (v) graphs are parallel straight lines, cutting frequency axis at different points (Fig.) This shows that threshold frequencies are different for different metals, the slope ( \dfrac{V_S}{v} ) for all the metals is same and hence universal constant.

(iv) Effect of time: There is no time lag between incidence of light and the emission of photoelectrons.

 

Einstein’s Explanation of Photo-electric Effect

 

The wave theory of light could not explain the observed characteristics of photo-electric effect. Einstein extended Planck’s quantum idea for light to explain photo-electric effect.

According to his idea, “The energy of electromagnetic radiation is not continuously distributed over the wave front like the energy of water waves but remains concentrated in packets of energy content hv, where v is frequency of radiations and h is universal Planck’s constant ( = 6.625 \times 10^{-34} j -s . Each packet of energy is called a photon or quantum and travels with the speed of light.

The assumptions of Einstein’s theory are :

1. The photo electric effect is the result of collision of two particles-one a photon of incident light and the other an electron of photo-metal.

2. The electron of photo-metal is bound with the nucleus by coulomb attractive forces. The minimum energy required to free an electron from its bondage is called work function (W).

3. The incident photon interacts with a single electron and loses its energy in two parts:

(i) in releasing the electron from its bondage, and

(ii) in impairing kinetic energy to emitted electron.

Accordingly if hv is the energy of incident photon, then

hv = W + E_k

 

Or \, \, \, E_k = \dfrac{1}{2} m v^2 _max = hv - W \cdots \, \, \, Equation \, \, \, 1

 

Where ‘W’ is work function and E_k = \dfrac{1}{2} m v^2 _{max} is the maximum kinetic energy of photo-electron.

Equation (1) is referred as Einstein’s photoelectric equation and explains all experimental results of photo-electric effect.

The efficiency of photo-electric effect is less than 1% of photons are capable of ejecting photo-electrons.

 

Photocells

 

A photocell is a device for covering light energy into electrical energy.

There are three main types of photocells:

(i) Photo emissive cells: In vacuum photo emissive cell, current is directly proportional to intensity.

(ii) Photo-voltaic cells

(iii) Photo-conductive cells

Photocells

Photocells


 

De-Broglie Waves

 

Light exhibits particle aspects in certain phenomena (e.g. photoelectric effect, emission and absorption of radiation) while wave aspects in other phenomena (e.g. interference, diffraction and polarization). That is light has dual nature. In analogy with dual nature of light, Louis de Broglie postulated that the material particles (e.g. electrons, protons, \alpha -particles, atoms etc.) may exhibit wave aspect. The wavelength associated with material particle having momentum p (mass m moving with velocity v) is given by:

\gamma = \dfrac{h}{p} = \dfrac{h}{mv}

 

If E_K is kinetic energy of moving material particle, then p = \sqrt{2m E_K}

 

\gamma = \dfrac{h}{\sqrt{2m E_K}}

I.e.

\gamma = \dfrac{h}{p} = \dfrac{h}{mv} = \dfrac{h}{\sqrt{2m E_k}}

 

The wave associated with material particle is called the de-Broglie wave. The de-Broglie hypothesis has been confirmed by diffraction experiments.

For charged particles accelerated through a potential difference of V – volts,

E_K = q V

 

\gamma = \dfrac{h}{\sqrt{2m q V}}

 

For electrons ( q = e = 1.6 \times 10^{-19} coul \, \, m = 9 \times 10^{-31} \, kg

 

\gamma = \dfrac{h}{\sqrt{2meV}} = \dfrac{6.62 \times 10^{-34}}{\sqrt{2 \times 9 \times 10^{-31} \times 1.6 \times 10^{-19} V}}

 

= \dfrac{12.27}{\sqrt{V}} \times 10^{-10} m = \dfrac{12.27}{\sqrt{V}} ]]> http://oscience.info/physics/electrons-and-photons/feed/ 0