LaboratoryExperimentsinOptics - site.iugaza.edu.pssite.iugaza.edu.ps/helegla/files/2010/02/optics.lab.tex.pdf · IslamicUniversity-Gaza FacultyofScience DepartmentofPhysics LaboratoryExperimentsinOptics

Islamic University-Gaza

Faculty of Science

Department of Physics

Laboratory Experiments in Optics

Prepared by

Zaher Nassar

and

Revised by

Dr. Hussian Dawoud

2007

Contents

1 Michelson Interferometer 6

1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Principle and Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Coherent sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 Composition of two simple harmonic motions in a straight line . . . 7

1.3.3 Michelson interferometer . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Setup and Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5.1 Adjustment of the interferometer . . . . . . . . . . . . . . . . . . . 11

1.5.2 Measuring the wavelength of the laser light . . . . . . . . . . . . . . 12

1.6 Measurements and Calculations . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Interference of Microwaves 14

2.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14


2.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Standing waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Interference of microwaves by using Michelson interferometer . . . . 16

2.4 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17


2.5.1 Reflection at the metal plate (standing waves) . . . . . . . . . . . . 18

2.5.2 Interference of Microwaves in the Michelson Interferometer . . . . . 19

3 Diffraction of Microwaves 21

3.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21


3.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Fresnel’s assumptions and Fresnel’s zones . . . . . . . . . . . . . . . 21

3.3.2 Huygens’ principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.3 Fresnel and Fraunhofer diffraction . . . . . . . . . . . . . . . . . . . 23

3.3.4 Fraunhofer diffraction at a single slit . . . . . . . . . . . . . . . . . 24

2

3.4 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27


3.6 Measurements and calculations . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Refraction Through Prism 30

4.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30


4.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3.1 Refraction Through a Prism . . . . . . . . . . . . . . . . . . . . . . 31

4.3.2 Minimum Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.3 The Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.4 Measurement of the angle A, of a prism . . . . . . . . . . . . . . . . 33

4.3.5 Measurement of the minimum deviation Dm . . . . . . . . . . . . . 34

4.4 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.5 Set up and procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.5.1 Determination of the refracting angle A . . . . . . . . . . . . . . . . 35

4.5.2 Determination of the angle of minimum deviation Dm . . . . . . . . 36


5 Polarimetry 38

5.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38


5.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.3.1 Polarization of light waves . . . . . . . . . . . . . . . . . . . . . . . 38

5.3.2 Optical activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3.3 Specific rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3.4 Laurent’s polarimeter . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.4 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42



6 Measuring the Velocity of Light 46

6.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46


3

6.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.3.1 Composition of two simple harmonic motions acting at right angles 47

6.3.2 Velocity of light in air . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.3.3 Velocity of light in medium . . . . . . . . . . . . . . . . . . . . . . 50

6.4 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


6.5.1 Adjustment of the apparatus . . . . . . . . . . . . . . . . . . . . . . 52

6.5.2 Measuring the velocity of light in air . . . . . . . . . . . . . . . . . 52

6.5.3 Measuring the velocity of light in medium . . . . . . . . . . . . . . 53

7 Newton’s Rings 55

7.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55


7.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.3.1 Newton’s rings by transmitted light . . . . . . . . . . . . . . . . . . 56

7.4 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58



8 Specific charge of the electron - e/me 60

8.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60


8.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

8.3.1 Deflection of the electron in magnetic field . . . . . . . . . . . . . . 60

8.3.2 Magnetic field on the axis of a circular coil . . . . . . . . . . . . . . 62

8.3.3 Helmholtz coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

8.4 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63



9 Diffraction grating 67

9.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67


9.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4

9.4 Fraunhofer diffraction at double slit . . . . . . . . . . . . . . . . . . . . . . 67

9.4.1 Interference maxima and minima . . . . . . . . . . . . . . . . . . . 68

9.4.2 Diffraction maxima and minima . . . . . . . . . . . . . . . . . . . . 70

9.4.3 Plane transmission diffraction grating . . . . . . . . . . . . . . . . . 70

9.4.4 Determination of the wave length of a spectral line using plane

transmission grating . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.5 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71



10 Laws of lenses 75

10.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


10.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

10.3.1 Thin Lens Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 75

10.3.2 Bessel’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

10.4 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77


10.5.1 Determination of the focal length of an unknown convex lens by

using thin lens equation method . . . . . . . . . . . . . . . . . . . . 78

10.5.2 Determination of the focal length of a convex lens by using Bessel’s

method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

10.6 Measurements and Calculations . . . . . . . . . . . . . . . . . . . . . . . . 79

5

1 Michelson Interferometer

1.1 Objective

Determination of the wave length of the light of the helium-neon laser by means of Michel-

son interferometer.

1.2 Principle and Task

Light is made to produce interference in the Michelson arrangement by the use of two

mirrors. The wave length is determined by displacing one mirror using the micrometer

screw.

1.3 Theory

The phenomenon of interference of light has proved the validity of the wave theory of

light. Interference is produced due to the superposition of two waves. It is not possible

to show interference due to two independent sources of light because a large number

of difficulties are involved. The two independent sources may emit light waves of largely

different amplitude and wavelength, and the phase difference between the two may change

with time.

1.3.1 Coherent sources

Two sources are said to be coherent if they emit light waves of the same frequency, nearly

the same amplitude and are always in phase with each other. It means that the two

sources must emit radiation of the same color (wavelength). In actual it is not possible

to have two independent sources which are coherent. But for experimental purpose, two

virtual sources formed from a single source can act as coherent sources. Interference of

light takes place between the waves by two methods:

1. From the real source and a virtual source.

2. From two virtual sources formed due to a single source.

In all such cases, the two sources will act, as if they are perfectly similar in all respects.

6

1.3.2 Composition of two simple harmonic motions in a straight line

Let two waves of the same frequency ω but of different amplitudes and different phase are

represented by the equations

y1 = a1 sin(ωt− α1) (1.1)

and

y2 = a2 sin(ωt− α2) (1.2)

If these two waves interfere, the resulting wave

y = y1 + y2 (1.3)

can be described as

y = A sin(ωt− α) (1.4)

with the amplitude

A2 = a21 + a2

2 + 2a1a2 cos δ, (1.5)

where

δ = α1 − α2 (1.6)

is the phase difference between the two waves.

For a1 = a2 = a

A2 = 4a2 cos2δ

2, (1.7)

The intensity distribution according to Eq. (1.7) is:

I ∼ A2 = 4a2 cos2δ

2(1.8)

Special cases

1. Maxima occur if the phase difference δ is an even of π

δ = 2nπ, n = 0, 1, 2, 3, · · · . (1.9)

2. Minima occur if the phase difference δ is an odd of π

δ = (2n+ 1)π, n = 0, 1, 2, 3, · · · . (1.10)

7

1.3.3 Michelson interferometer

Michelson interferometer consists of two highly polished mirrors M1 and M2 and two

plane glass plates A and C parallel to each other. The rear side of the glass plate A is

half silvered so that light coming from the source S is equally reflected and transmitted

by it. Light from a monochromatic source S after passing through the lens L, falls on the

plate A. The lens L makes the beam parallel. The plate A is inclined at an angle of 450.

One-half of the energy of the incident beam is reflected by the plate A towards the mirror

M1 and the other half is transmitted towards the mirror M2. These two beams (reflected

and transmitted) are reflected back by the mirrors M1 and M2. These two beams return

to the plate A. The beam reflected back by M1 is transmitted through the glass plate

A and the beam reflected back by M2 is reflected by glass plate A towards the eye as

shown in Fig. 1.1. The beam going towards the mirror M1 and reflected back, has to

Figure 1.1: Michelson interferometer.

8

pass twice through the glass plate A. Therefore, to compensate for the path, the plate C

is used between the mirror M2 and A. The light beam going towards the mirror M2 and

reflected back towards A also passes twice through the compensating plate C. Therefore,

the paths of the two rays in glass are the same. The mirror M2 can be moved with the

help of the micrometer handle H. The distance through which the mirror M2 is moved

can be read on the scale of the micrometer. The planes of the mirrors M1 and M2 can be

made perfectly perpendicular with the help of the fine screws attached to them.

• Remark

The compensating plate C is a necessity for white light fringes but can be dispensed with,

while using monochromatic light.

If the mirrors M1 and M2 are perfectly perpendicular, the observer’s eye will see the

images of the mirrors M1 and M2 through A. There will be an air film between the two

images and distance can be varied with the help of the micrometer handle H. The fringes

will be perfectly circular. If the two images ofM1 andM2 are inclined (the mirrorsM1 and

M2 not perfectly perpendicular) the enclosed air film will be wedge-shaped and straight

line fringes will be observed. When the mirror M2 is moved away or towards the glass

plate A with the help of handle H, the fringes cross the center of the field of view of the

observer’s eye. If M2 is moved through a distance λ/2, one fringe will cross the field of

view and will move to the position previously occupied by the next fringe.

Circular fringes are produced in Michelson interferometer if the two mirrors M1 and M2

are perpendicular to each other, that is, if the mirror M1 is parallel to the virtual mirror

M′

2 which is the image of M2 (Fig. 1.2).

The source is an extended one and S1 and S2 are the virtual images of the source due to

M1 and M′

2. If the distance M1M′

2 is d, the distance between S1 and S2 is 2d. The path

difference between the two beams will be

Γ = 2d cos θ. (1.11)

The relation between the phase difference δ and the path difference Γ is

δ =2π

λΓ. (1.12)

Substituting from Eq. (1.9) and Eq. (1.11) into Eq. (1.12), maxima thus occur if

2d cos θ = nλ, n = 0, 1, 2, 3, · · · . (1.13)

9

Figure 1.2: Formation of circles on interference.

If the position of the movable mirror M2 is changed so that d for example decreases then,

according to Eq. (1.13), the diameter of the ring will also decrease since n is fixed for this

ring:

d ↓⇒ cos θ ↑⇒ θ ↓⇒ radius of the nth ring ↓ .

For the first bright ring (n = 1), if this ring is disappeared (θ = 0) then, from Eq. (1.13)

d =λ

2. (1.14)

One ring thus disappears each time d is reduced by λ/2. If the movable mirror is displaced

a distance d′

(d′

is the micrometer reading), a number n of fringes is crossed the field of

view.

Ifλ

2∼ 1

d′

∼ n,

10

then

d′

= nλ

2(1.15)

or

2d′ = nλ. (1.16)

1.4 Equipment

Support base, helium-neon lazer, Michelson interferometer, screen.

1.5 Setup and Procedure

The experimental set up is as shown in Fig.1.3. In order to obtain the largest possi-

Figure 1.3: Experimental set-up for measuring wavelengths with the Michelson in-

terferometer.

ble number of interference fringes, the two mirrors of the interferometer are first of all

adjusted.

1.5.1 Adjustment of the interferometer

1. The laser beam is adjusted to strike the half-silvered mirror at an angle of 45◦.

11

2. The laser beam is split into two beams.

3. The resulting two beams are reflected by the mirrors and impinge on the screen.

4. Two points are obtained on the screen.

5. By means of the two adjusting screws, both points of light are made to coincide.

6. The points of light are enlarged and interference patterns are observed on the screen

(bands, circles).

7. By careful readjustment, an interference image of concentric circles can be obtained.

1.5.2 Measuring the wavelength of the laser light

1. The micrometer screw is turned to any initial position at which the center of the

circles is dark.

2. The initial reading of the micrometer is read.

3. The micrometer screws is now further turned in the same direction and 40 light-dark

periods are counted.

4. The distance travelled by the mirror must be read on the micrometer screw and

divided by ten (when the micrometer screw moves 10 mm the mirror is displaced

1 mm ).

5. If the central point of the circles moves outside the light spot area, a readjustment

has to be performed in order to obtain concentric fringes.

17. Repeat paragraphs 13-16 for 60, 80, 100, 120, 140,160 and 180 circles.

18. Record your measurements in Table 1.1.

1.6 Measurements and Calculations

• Plot n as x-axis versus d′

as y-axis.

• Find the slope, then calculate the wavelength λ from Eq. (1.15) where

λ = 2 slope.

12

Table 1.1: The results of a typical measurements for the determination of the wave

length of the light of the helium-neon laser by means of Michelson interferometer.

n 40 60 80 100 120 140 160 180

d′

(mm)

Figure 1.4: Displacement of the mirror d as a function of number of fringes n that

crossed the field of view in Michelson interferometer.

13

2 Interference of Microwaves

2.1 Objective

Measurement of the wavelength of microwaves through the production of standing waves

with

1. reflection at the metal screen.

2. the Michelson interferometer.


If a microwave beam is incident on a metal screen or glass plate, it will reflect and

interferes with the primary waves. The result of interference is said to be standing wave.

The wavelength of the microwave is determined from the resultant standing waves.

2.3 Theory

Interference of microwaves can take place by two methods:

1. Due to the superposition of the incident wave and reflected wave (standing waves).

2. Due to the superposition of two coherent virtual waves (Michelson interferometer).

2.3.1 Standing waves

Electromagnetic waves can be reflected; a conducting surface can serve as a reflector. The

superposition principle holds for electromagnetic waves just as for electric and magnetic

fields. The superposition of an incident wave and a perfectly reflected wave forms a

standing wave. The situation is analogous to standing waves on a stretched string.

We can derive a wave function for the standing wave by adding the functions for two waves

with equal amplitude, period and wave length, traveling in opposite directions. Call these

two wave functions E1 and E2.

Suppose E1 is the incident wave traveling to the left and described by the wave function

E1 = A sin(ωt+ kx), (2.1)

14

and E2 is the reflected wave traveling to the right and described by the wave function

E2 = −A sin(ωt− kx), (2.2)

where A is the amplitude of the wave, ω = 2πf is the angular frequency and k = 2π/λ is

the propagation constant.

These two waves interfere with each other, and the total field at any point, according to

Figure 2.1: Intensity distribution of the reflection of microwaves, as a function of

the distance from the reflecting plate.

superposition principle, is given by

E = E1 + E2 = A {sin(ωt+ kx)− sin(ωt− kx)} . (2.3)

We can expand and simplify this expression, using the identities

sin(A±B) = sinA cosB ± cosA sinB. (2.4)

The result is

E = (2A cosωt) sin kx. (2.5)

E thus disappears for

kx = nπ, n = 0, 1, 2, 3 · · · . (2.6)

15

Since

k =2π

λ, (2.7)

then

x = n ·λ

2, n = 1, 2, 3 · · · (2.8)

From the distance between the maxima in Fig. 2.1, the wavelength can be calculated

using Eq. (2.8).

2.3.2 Interference of microwaves by using Michelson interferometer

Suppose a microwave beam, emitted from the source S, is incident on the glass plate G

as shown in Fig. 2.2. One half of the energy of the incident beam is reflected by the glass

Figure 2.2: Michelson arrangement for the interference of microwaves.

plate G towards the metal plate P1 and the other half is transmitted towards the metal

plate P2. These two beams (reflected and transmitted) are reflected back by the plates P1

and P2 to return to the plate G. The beam reflected back by P1 is transmitted through

the glass plate G and the beam reflected back by P2 is reflected by glass plate towards

16

the receiver R. Interfere will take place between these two coherent waves. When one of

the two plates P1 or P2 is moved towards the plate G, minima and maxima intensities are

formed as a function of the displacement of the plate as shown in Fig. 2.3. If the movable

Figure 2.3: Intensity distribution during interference of microwaves in the Michelson

arrangement, as a function of the displacement of the reflection plate.

plate is displaced by λ/2, the intensity will disappear (one minimum is obtained) and if

it is moved a distance x, a number n of minima is produced, that is if

λ

2∼ 1 (2.9)

and

x ∼ n, (2.10)

then

x = nλ

2. (2.11)

Thus the wavelength can be obtained by measuring the distances between the maxima.

2.4 Equipment

Microwave transmitter, microwave receiver, microwave receiving dipole, microwave power

supply, protractor scale with pointer, glass plate, two metal plates, plate holders, clamps,

17

meter scale, tripod base, support rod, multimeter and connecting cable.


2.5.1 Reflection at the metal plate (standing waves)

The experimental set up for reflection at the metal plate is shown in Fig. 2.4. The

microwave receiving dipole is used for field-strength measurement.

Figure 2.4: Experimental set up for interference of microwaves with a plane metal

plate.

1. The experimental set up for reflection at the metal plate is arranged (microwave

source, microwave receiving dipole and metal plate). The receiving dipole is placed

on a meter stick.

2. Consider any initial position for the dipole(x = 0).

3. Record the multimeter reading for x = 0 (the multimeter is adjusted to measure DC

voltage in mV ).

4. Change the position of the dipole by 0.5 cm and record the multimeter reading.

5. Repeat paragraph (4) for forty readings.


18

7. Plot the voltage V (mv) as y-axis versus the displacement x(cm) as x axis.

8. Measure the distance x between two maxima and count the number of minima n

within the distance x as shown in Fig. 2.1.

9. calculate the wave length of the microwave from Eq. (2.8).


length of the microwave by means of reflection on the metal plate.

x(cm) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 .... .... .... .... 20

V (mv)

2.5.2 Interference of Microwaves in the Michelson Interferometer

The experimental set up for the Michelson interferometer is shown in Fig. 2.2 and

in Fig. 2.5.

Figure 2.5: Experimental set up for the interference of microwaves in the Michelson

interferometer (beneath).

1. The Michelson interferometer set up is arranged as shown in Fig. 2.5.

2. The two metal plates are placed at equal distances from the glass plate.

19

3. Consider the position of one of the metal plates to have the coordinate x = 0.

4. Record the multimeter reading for x = 0.

5. Displace the metal plate by 0.5 cm and record the multimeter reading.

6. Repeat paragraph (5) for forty readings.


8. Plot the voltage V (mv) as y-axis versus the displacement x(cm) as x axis.

9. Measure the distance x between two maxima and count the number of minima n

within the distance x as shown in Fig. 2.1.

10. calculate the wave length of the microwave from Eq. (2.11).


length of the microwave by means of Michelson interferometer.

x(cm) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 .... .... .... .... 20

V (mv)

20

3 Diffraction of Microwaves

3.1 Objective

Determination of the diffraction pattern of the microwave intensity after passing through

a slit.


Microwaves impinge on a slit. The diffraction pattern is determined on the basis of

diffraction at this slit.

3.3 Theory

If an opaque obstacle is placed in the path of light, a sharp shadow is cast on the screen,

indicating thereby that light travels in straight lines. But it has observed that when a

beam of light passes through a small opening (a small circular hole or a narrow slit) it

spreads to some extent into the region of the geometrical shadow also. If light energy is

propagated in the form of waves, then similar to sound waves one would expect bending of

the beam of light round the edges of an opaque obstacle or illumination of the geometrical

shadow. The bending of light round the edges of an obstacle or the encroachment of light

within the geometrical shadow is called diffraction.

3.3.1 Fresnel’s assumptions and Fresnel’s zones

According to Fresnel, the resultant effect at an external point due to a wavefront will

depend on the factors discussed below:

In Fig. 3.1, S is a point source of monochromatic light, MN is a small aperture and XY

is the screen. MCN is the incident spherical wave front due to the point source S. To

obtain the resultant effect at a point P on the screen, Fresnel assumed that:

1. A wavefront can be divided into a large number of strips or zones called Fresnel’s zones

of small area and the resultant effect at any point will depend on the combined effect

of all the secondary waves emanating from the various zones.

2. the effect at a point due to any particular zone will depend on the distance of the point

from the zone.

21

Figure 3.1: Fresnel’s assumptions and Fresnel’s zones

3. The effect at P will also depend on the obliquity of the point with reference to the

zone under consideration, e.g. due to the part of the wavefront at C, the effect will

be maximum at O and decreases with increasing obliquity. The effect at a point

due to the obliquity factor is proportional to (1 + cos θ).

Now suppose that ABCD is a plane wave front perpendicular to the plane of the paper

(Fig. 3.2) and P is an external point at a distance d perpendicular to ABCD. To find

the resultant intensity at P due to the wavefront ABCD, Fresnel’s method consists in

dividing the wavefront into a number of half period elements or zones called Fresnel’s

zones.

With P as center and radii equal to b+ λ2, b+ 2λ

2, b+ 3λ

2· · · etc., construct spheres which

will cut out circular areas of radii OM1, OM2, OM3 · · · etc., on the wavefront. These

circular zones are called half period zones or half period elements. Each zone differs from

its neighbor by a phase difference of π or a path difference of λ2. Thus the secondary waves

starting from the points O and M1 and reaching P will have a phase difference of π or a

path difference of λ2.

3.3.2 Huygens’ principle

Huygens assumed that every point of a wave front may be considered the source of sec-

ondary waves that spread out in all directions with a speed equal to the speed of propaga-

tion of the wave. The new wavefront at a later time is then found by constructing a surface

22

Figure 3.2: Fresnel’s method to find the resultant intensity at point P due to plane

wavefront.

tangent to the secondary waves, which is called the envelope of the wavelets. Huygens’

principle is illustrated in Fig 3.3. The original wavefront AA′

is traveling outward from a

source S. Every point lies on the wavefront AA′

represents a secondary source, and the

envelope of the secondary wavelets constructs the new wavefront BB′

.

3.3.3 Fresnel and Fraunhofer diffraction

Diffraction phenomena can conveniently be divided into two groups viz., (1)Fresnel diffrac-

tion phenomena and (2)Fraunhofer diffraction phenomena.

1. Fresnel diffraction phenomena

In the Fresnel class of diffraction, the source or the screen or both are at finite

distances from the aperture or obstacle causing diffraction. In such a case, the

phenomenon observed on the screen is called Fresnel diffraction pattern.

2. Fraunhofer diffraction phenomena

23

Figure 3.3: Applying Huygens’ principle to wave front AA′to construct a new wave

front BB′.

In the Fraunhofer class of diffraction phenomena, the source and the screen on which

the pattern is observed are at infinite distances from the aperture or the obstacle

causing diffraction.

3.3.4 Fraunhofer diffraction at a single slit

To obtain Fraunhofer diffraction pattern, the incident wavefront must be plane and the

diffracted light is collected on the screen with the help of a lens. Thus the source of light

should either be at a large distance from the slit or a collimating lens must be used.

In Fig. 3.4, S is a narrow slit perpendicular to the plane of the paper and illuminated

by monochromatic light. L1 is the collimating lens and AB is a slit of width w. XY is

the incident spherical wavefront. The light passing through the slit AB is incident on the

lens L2 and the final refracted beam is observed on the screen MN .

A plane wavefront is incident on the slit AB and each point on this wavefront is a

source of secondary waves traveling in the direction parallel to OP viz. AQ and BV

come to focus at P and a bright central image is observed. The secondary waves from

points equidistant from O and situated in the upper and lower halves OA and OB of the

wavefront travel the same distance in reaching P and hence the path difference is zero.

24

Figure 3.4: Fraunhofer diffraction at a single slit.

The secondary waves reinforce one another and P will be a point maximum intensity.

Now consider the secondary waves traveling in the direction AR, inclined at an angle θ to

the direction OP . All the secondary waves traveling in this direction reach the point P′

on the screen. The point P′

will be of maximum or minimum intensity depending on the

path difference between the secondary waves originating from the corresponding points

of the wavefront. Draw OC and BL perpendicular to AR. Then, in the 4ABL

sin θ =AL

AB=AL

w, (3.1)

or

Γ(= AL) = w sin θ, (3.2)

where w is the width of the slit and Γ(= AL) is the path difference between the secondary

waves originating from A and B.

Radiation intensity produced at P′

If the primary waves arriving at the slit and the waves measured at P′

can be regarded

approximately as plane waves (Fraunhofer diffraction), the radiation intensity produced

at P′

is

I = I0

(

sin(δ/2)

(δ/2)

)2

, (3.3)

where I0 is the intensity at the point P and δ is the phase difference between the secondary

waves.

Special cases:

25

1. Intensity is minimum when the phase difference δ is an even number multiple of π,

i.e., when

δ = 2nπ, n = 1, 2, 3 · · · . (3.4)

∵ δ =2π

λΓ, (3.5)

2nπ =2π

λΓ, (3.6)

∴ Γ = nλ. (3.7)

Thus, destructive interference takes place when the path difference is an integer

multiple of λ, and the point P′

will be of minimum intensity.

Equating Eq. (3.2) and Eq. (3.7) we can write

sin θn =nλ

w, n = 1, 2, 3 · · · , (3.8)

where θn gives the direction of the nth minimum.

2. Intensity is maximum when the phase difference δ is an odd number multiple of π, i.e.,

when

δ = (2n+ 1)π, n = 1, 2, 3 · · · . (3.9)

Similarly, we can show that a constructive interference takes place when the path

difference is an odd multiple of λ2, i.e., when

Γ =(2n+ 1)

2λ, n = 1, 2, 3 · · · , (3.10)

and the point P′

will be of maximum intensity.

Equating Eq. (3.2) and Eq. (3.10) gives

sin θn =(2n+ 1)λ

2w, n = 1, 2, 3 · · · , (3.11)

where θn gives the direction of the nth maximum.

Thus, the diffraction pattern due to a single slit consists of a central bright maximum at

P followed by secondary maxima and minima on both the sides as shown in Fig. 3.5.

If the lens L2 is very near the slit or the screen is far away from the lens L2, then

sin θn =xnf

=xnd, (3.12)

26

Figure 3.5: The intensity distribution in the diffraction of waves at the slit as a

function of the position along a straight line parallel to the plane of the slit.

where d is the distance between the slit and the screen. From Eq. (3.8) and Eq. (3.12),

we can writexnd

=nλ

w, (3.13)

or

xn =nλd

w, n = 1, 2, 3 · · · (3.14)

where xn is the distance of the nth minimum from the central point P .

Similarly, from Eq. (3.11) and Eq. (3.12), we can show that

xn =(2n+ 1)λd

2w, n = 1, 2, 3 · · · (3.15)

where xn is the distance of the nth maximum from the central point P .

3.4 Equipment

Microwave transmitter, microwave receiving dipole, microwave power supply, two metal

plates, plate holders, clamps, meter scale, tripod base, support rod, multimeter and con-

necting cable.

27


The experimental set up is as shown in Fig. 3.6. The microwaves (9.45 GHz) used

for detection are amplitude-modulated either internally with a frequency of 50 Hz or

externally with any frequency(NF). Two metal screens, the edges of which are 6 cm

apart, form a slit. The microwave beam impinges at right angles on to the slit and is

adjusted parallel to the screens using the receiving diode. The receiving dipole is moved

parallel to the screens.

Figure 3.6: Experimental set up for determining the diffraction pattern of mi-

crowaves at the slit.

1. A distance of approximately 80cm from the transmitter to the slit, and approximately

20cm from the slit to the receiver, is recommended.

2. Adjust the slit width to 6cm.

3. Put the receiver at the position of the central maximum as shown in Fig. 3.6.

4. The position of the transmitter is adjusted so that the reading of the multimeter, at

the central point(x = 0), is maximum value.

5. Record the multimeter reading for x = 0.

28

6. Change the position of the dipole by 0.5 cm to the right(in the +ve. x direction) and

record the multimeter reading.

7. Repeat paragraph 6, step by step, forty times.

8. Repeat paragraphs 6 and 7 for negative values of x.


3.6 Measurements and calculations

Table 3.1: The results of a typical measurements for the determination of the diffrac-

tion pattern of the microwave intensity after passing through a slit.

x(cm) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 .... .... .... .... 20

V (mv)

x(cm) 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 .... .... .... .... -20

V (mv)

(a) Plot V (mv) as y-axis versus x(cm) as x-axis.

(b) Measure xn for the 1st and 2nd maxima in the two directions and calculate the wave

length of the microwave from Eq. (3.15), where

w = slit width.

d= slit-receiver separation.

n= fringe order.

(c) As in step (b) calculate λ, for the dark fringes, using Eq. (3.14).

(c) Compute the average value of the wave length of microwave.

29

4 Refraction Through Prism

4.1 Objective

1.To determine the refractive index of glass prism.

2.To determine the refractive index of various liquids in a hollow prism.


The refractive indices of liquids and glass are determined by refraction of light through

the prism at minimum deviation.

4.3 Theory

A glass prism is a transparent object which has two plane surfaces, XDEY and XDFZ,

inclined to each other (Fig. 4.1).

The angle between the inclined plane surfaces XDFZ and XDEY is known as the

Figure 4.1: Prism.

angle of the prism, or the refracting angle, the line of the intersection XD of the planes is

known as the refracting edge. A ray of light ab, incident on the prism at b in a direction

perpendicular to XD, is refracted towards the normal along bc when it enters the prism,

and is refracted away from the normal along cd when it emerges into the air. If the

incident ray is directed towards the refracting angle, as in Fig. 4.1, the emersion light is

always deviated by the prism towards its base.

30

4.3.1 Refraction Through a Prism

Consider a ray HM incident on a prism of refracting angle A ( Fig. 4.2). If i1, r1 and i2, r2

are the angles of incidence and refraction at M and N as shown, and n is the refractive

index of the prism then, by using the law of refraction,

sin i1 = n sin r1, (4.1)

sin i2 = n sin r2. (4.2)

From the geometry of Fig. 4.2 we can write:

Figure 4.2: Refraction through prism.

∴ A = r1 + r2, (4.3)

as angle NST is the exterior angle of the triangle MSN .

The deviation angle, D, of the light (angle BOK), is given by

D = (i1 − r1) + (i2 − r2). (4.4)

Equations (4.1) - (4.3.3) are the general relations which hold for refraction through a

prism.

4.3.2 Minimum Deviation

Experiment shows that as the angle of incidence i is increased from zero, the deviation

D begins to decrease continuously to some value Dm, and then increases to a maximum

31

Figure 4.3: Minimum deviation.

as i is increased further to 900. A graph of D plotted against i has the appearance of the

curve X, which has a minimum value at R, Fig. 4.3. Experiment and theory show that

the minimum deviation, Dm, of the light occurs when i1 = i2 = i and r1 = r2 = r .

Then, we can write Eqs. (4.1), (4.3) and (4.4) as

n =sin i

sin r, (4.5)

A = 2r ⇒ r =A

2, (4.6)

Dm = 2i− 2r = 2i− A ⇒ i =Dm + A

2. (4.7)

Substituting Eqs. (4.3.5) and (4.5) into Eq. (4.3.4) gets the refractive index of the prism

n =sin Dm+A

2

sin A2

. (4.8)

4.3.3 The Spectrometer

The spectrometer is an optical instrument used to measure accurately the refractive index

of glass in the form of prism. It consists essentially of a collimator C, a telescope T , and

a table R, on which a prism B can be placed. The collimator is fixed, but the table and

the telescope can be rotated round a circular scale (Fig. 4.4). The collimator is used to

32

obtain parallel rays, and the telescope to brought these rays to a focus at cross-wires. A

vernier is also provided for this scale. The source of light S, used in the experiment is

placed in front of a narrow slit at one end of the collimator.

Figure 4.4: Spectrometer.

Before the spectrometer can be used, however, three adjustments must be made:

(1) The collimator C must be adjusted so that parallel light emerges from it.

(2) The telescope T must be adjusted so that parallel rays entering it are brought to a

focus at cross-wires near its eye-piece.

(3) The refracting edge of the prism (which is the edge XD in Fig. 4.1) must be parallel

to the axis of rotation of the telescope (this axis is perpendicular to the plane of the

parer in Fig. 4.5), i.e., the table must be ”levelled”.

4.3.4 Measurement of the angle A, of a prism

The prism is placed so as to face the collimator lens. The incident light illuminates the

two surfaces containing the refracting angle A with parallel light as in Fig. 4.5. The

parallel rays will reflect, and an image of the collimator slit is hence observed with the

telescope in positions T1, T2. The angle of rotation of the telescope from T1 to T2 is equal

to A1 + A2 = 2A, and hence the angle of the prism, A, can be obtained.

• Using the geometry of Fig. 4.5 show that

A(= α+ β) =A1 + A2

2

33

Figure 4.5: Measurement of angle of prism.

4.3.5 Measurement of the minimum deviation Dm

To measure the minimum deviation Dm, the prism is placed with its refracting angle A

pointing away from the collimator, as shown in Fig. 4.6.The telescope is then turned

until an image of the slit is obtained on the cross-wires at the position T1. The table is

now slowly rotated so that the angle of incidence decreases, and the image of the slit is

kept on the cross-wires by moving the telescope at the same time. The image of the slit,

and the telescope, then slowly approach the fixed line XY . But at one position (T2), the

image of the slit begins to move away from XY . If the table is now turned in the opposite

direction the image of the slit again moves back when the telescope reaches the position

T2. The angle between the emergent ray CH and the line XY is hence the smallest angle

of deviation, and is thus equal to Dm.

4.4 Equipment

Spectrometer, lamp holder, Spectral lamp Hg, Power supply for spectral lamps, glass

Prism , hollow prism filled with paraffin and hollow prism filled with glycerin.

34

Figure 4.6: Measurement of minimum deviation.

4.5 Set up and procedure

The experiment is set up as shown in Fig. 4.7. The spectrometer and the prism should

be adjusted in accordance with the operating instructions.

The aperture, or slit, is projected into the plane of the cross-wires with the telescope set

to infinity and observed with the eyepiece which is used as a magnifier.

4.5.1 Determination of the refracting angle A

1. Place the prism on the table of the spectrometer with the angle to be measured

directed toward the lens of the collimator.

35

Figure 4.7: Experimental set-up for determining dispersion in prism.

2. Turn the telescope till you get an image formed by reflection on the surface of the

prism.

3. Determine the position of the telescope (A1) by reading the scale.

4. Turn the telescope in the opposite direction to get another image of the slit reflected

on the second face of the prism, and determine the position (A2) in this case.

4.5.2 Determination of the angle of minimum deviation Dm

1. Place the glass prism on the table so that the angle (A) which was measured serves

as the refracting angle.

2. Turn the telescope till you gets spectral lines and measure the minimum deviation

angle (Dm) for each spectral line, and record the results in Table 4.1.

• The position of minimum deviation can be detected by looking to the spectrum

through the telescope and rotating it until the spectrum reverse its direction.

2. Repeat the steps in sections 4.5.1 and 4.5.2, for the liquid prism, and record the

results in Table 4.2.

36


Table 4.1: Typical measurements for determining the refractive index of the glass

prism.

Yellow Green Blue Violet

Dm

n

Table 4.2: Typical measurements for determining the refractive index of the liquid

prism.


Dm

n

(a)In the first parts of the experiment, determine the angle A of the prism from the

relation:

A =A1 + A2

2.

(b) In the second part, Calculate the refractive index (n) for each spectrum from the

relation:

n =sin A+Dm

2

sin A2

.

37

5 Polarimetry

5.1 Objective

To determine the specific rotation of a sugar by measuring the rotation of various solutions

of known concentration.


The rotation of the plane of polarization through a sugar solution measured with a po-

larimeter and the specific rotation of the sugar determined.

5.3 Theory

The experiments on interference and diffraction have shown that light is a form of wave

motion. These effects do not tell us about the type of wave motion i.e., whether the

light waves are longitudinal or transverse, or whether the vibrations are linear, circular

or torsional. The phenomenon of polarization has helped to establish beyond doubt that

light waves are transverse waves.

5.3.1 Polarization of light waves

Let light from a source S fall on a tourmaline crystal A which is cut parallel to its axis

(Fig. 5.1). On rotating the crystal A, no remarkable change is noticed. Now place the

crystal B parallel to A.

1. Rotate the crystals together so that their axes are always parallel. No change is

observed in the light coming out of B (Fig. 5.1 a).

2. Keep the crystal A fixed and rotate the crystal B. The light transmitted through B

becomes dimmer and dimmer. When B is at right angles to A, no light emerges out

of B (Fig. 5.1 b).

If the crystal B is further rotated, the intensity of light coming out of it gradually in-

creases and is maximum again when the two crystals are parallel.

This experiment shows conclusively that light is not propagated as longitudinal or com-

pressional waves. If we consider the propagation of light as a longitudinal wave motion

38

Figure 5.1: Polarization of light.

then no extinction of light should occur when the crystal B is rotated.

It is clear that after passing through the crystal A, the light waves vibrate only in one

direction. Therefore light coming out of of the crystal A is said to be polarized because

it has acquired the property of one-sidedness with regard to the direction of the rays.

This experiment proves that light waves are transverse waves.

5.3.2 Optical activity

Optical activity is the ability of many substances to rotate the plane of vibration of a ray

of plane-polarized light passing through them. It is caused by the two components of a

plane polarized ray of light travelling at different speeds through the asymmetric medium

and, in so doing, undergoing a phase shift with respect to one another, this phase shift

being indicated as a rotation of the plane of polarization.

5.3.3 Specific rotation

Liquids containing an optically active substance (like sugar solution), rotate the plane of

the linearly polarized light. The angle through which the plane polarized light is rotated

depends upon:

1. the thickness of the medium.

2. concentration of the solution or density of the active substance in the solvent.

39

3. wave length of light.

4. temperature.

The specific rotation (rotatory power) is defined as the rotation produced by a decimeter

(10cm) long column of the liquid containing 1gm of the active substance in one cc (cm3)

of solution. Therefore,

Stλ =10θ

lc(5.1)

where Stλ represents the specific rotation at temperature t oC for a wave length λ,θ is the

angle of rotation, l is the length of the solution in cm through which the plane polarized

light passes and c is the concentration of the active substance ingm/cm3 in the solution.

The angle through which the plane of polarization is rotated by the optically active

substance is determined with the help of a polarimeter.

5.3.4 Laurent’s polarimeter

It consists of two nicol prisms N1 and N2 (Fig. 5.2). N1 is a polarizer and N2 is an

analyser. Behind N1, there is a half wave plate of quartz Q which covers one half of the

field of view, while the other half G is a glass plate. The glass plate G absorbs the same

amount of light as the quartz plate Q. T is a hollow glass tube having a large diameter at

its middle portion (T-shaped tube can be used). When this tube is filled with the solution

containing an optically active substance and closed at the ends by cover-slips and metal

covers, there will be no air bubbles in the path of light. The air bubbles (if any) will

appear at the upper portion of the wide bore T1 of the tube.

Light from a monochromatic source S is incident on the converging lens L. After pass-

Figure 5.2: Laurent’s half shade polarimeter.

40

ing through N1, the beam is plane polarized. One half of the beam passes through the

quartz plate Q and the other half passes through the glass plate G. Suppose the plane of

vibration of the plane polarized light incident on the half shade plate is along AB (Fig.

5.3). Here AB makes an angle θ with Y Y′

. On passing through the quartz plate Q, the

Figure 5.3: Direction of vibration of the plane polarized light before and after passing

the half shade plate.

beam is split up into ordinary and extraordinary components which travel along the same

direction but with different speeds and on emergence a phase difference of π or a path

difference of λ/2 is introduced between them. The vibration of the beam emerging out

of quartz will be along CD whereas the vibration of the beam emerging out of the glass

plate will be along AB. If the analyser N2 has its principal plane or section along Y Y′

i.e., along the direction which bisects the angle AOC, the amplitudes of light incident

on the analyser N2 from both the halves (i.e., quartz half and glass half) will be equal.

Therefore, the field of view will be equally bright (Fig. 5.4 a).

If the analyser N2 is rotated to the right of Y Y′

, then the right half will be brighter as

compared to the left half (Fig. 5.4 b). On other hand, if the analyser N2 is rotated to the

left of Y Y′

, the left half is brighter as compared to the right half (Fig. 5.4 c).

Therefore, to find the specific rotation of an optically active substance (say, sugar solu-

tion), the analyser N2 is set in the position for equal brightness of the field of view, first

without the solution on the tube T . The reading of vernier V1V2 is noted. When a tube

containing the solution of known concentration is placed, the vibrations from the quartz

half and the glass half are rotated. In the case of sugar solution, AB and CD are rotated

on the clockwise direction. Therefore, on the introduction of the tube containing the sugar

41

Figure 5.4: Brightness of the field of view.

solution, the field of view is not equally bright. The analyser is rotated in the clockwise

direction and is brought to a position so that the whole field of view is equally bright. The

new position of of the vernier V1V2 on the circular scale is read. Thus, the angle through

which the analyser has been rotated gives the angle through which the plane of vibration

of the incident beam has been rotated by the sugar solution. In the actual experiment,

for various concentrations of the sugar solution, the corresponding angles of rotation are

determined. A graph is plotted between concentration C and the angle of rotation θ. The

graph is a straight line.

Then from the relation

Stλ =10θ

lC,

the specific rotation of the optically active substance is calculated.

5.4 Equipment

Polarimeter, plastic beaker 250 ml, plastic graduated cylinder 100 ml, plastic graduated

vessel 1000 ml, glass rod, sugar, distilled water and balance.


To measure optical activity we use a polarimeter in which light of wave-length 589 nm

(sodium-D line) is first plane-polarized by a polarizer and then studied by a second po-

larizer (analyser) to find its new plane of polarization after passing through the sample

42

Figure 5.5: Experimental set-up for measuring the rotation of sugar solutions.

substance. The analyser is now set to equal brightness on the two sides of the field of

view, without the solution in the T-shaped tube shown in Fig. (5.5).

1. First prepare a sugar solution of known concentration (co = 0.24gm/cm3) by dis-

solving 12gm of the sugar and making up to a total volume of 50cm3 with distilled

water in the measuring cylinder.

2. The analyser is now set to equal brightness on the two sides of the field of view,

without the solution in the T-shaped tube.

3. Record the initial reading θo of the vernier scale.

4. Fill the T-shaped tube (the tube must be cleaned with distilled water) with the

solution, and introduce it in the ray path (the field of view is now not equally

bright).

5. Adjust the two halves of the field to identical brightness by using the fine screw of

the polarimeter.

6. Record the reading of the vernier scale, then find the angle θ through which the plane

polarized light has been rotated (θ = reading of the vernier scale± θo).

7. Four concentrations, co/2, co/4, co/8, co/16, are made by making up the solution re-

maining in the measuring cylinder each time with exactly the same volume of dis-

43

tilled water.

• Remark

Don’t prepare these solutions until you finish the measurement of the previous one,

that is, prepare co/2 then proceed to step 8 and so on.

8. Repeat paragraphes 4-6 for each of the solutions prepared in step 7, and record the

results in Table 5.1.


Table 5.1: The results of a typical measurements for the determination of the specific

rotation of the cane sugar.

c (gm/cm3) 0.24 0.12 0.06 0.03 0.015

θo

A graph is plotted between concentration c as x-axis and the angle of rotation θ as

y-axis (Fig. 5.6).

From the relation

Stλ =10θ

lc,

the specific rotation of sugar can be calculated.

Stλ =θ

c= slope,

(related to a column length of 10 cm)

44

Figure 5.6: Angle of rotation θ as a function of concentration c of the sugar.

45

6 Measuring the Velocity of Light

6.1 Objective

1. To determine the velocity of light in air.

2. To determine the velocity of light in water and synthetic resin and to calculate the

refractive indices.


The intensity of the light is modulated and the phase relationship of the transmitter and

receiver signal compared. The velocity of light is calculated from the relationship between

the changes in the phase and the light path.

Figure 6.1: Experimental set-up for vacuum measuring the velocity of light in syn-

thetic resin.

6.3 Theory

When waves travel in a homogeneous isotropic material (a material with the same prop-

erties in all regions and in all directions), the rays are always straight lines normal to

the wave fronts. At a boundary surface between two materials, such as the surface of a

glass plate in air, the wave speed and the direction of a ray usually change, but the ray

46

segments in the air and in the glass are straight lines. The ratio of the speed of light in

air ca to the speed of light in a medium cm is called the refractive index n of the medium.

n =cacm

. (6.1)

6.3.1 Composition of two simple harmonic motions acting at right angles

Let

x = a sin(ωt+ α) (6.2)

and

y = b sin(ωt) (6.3)

represent the displacements of a particle along the x− and y−axes due to the influence

of two simple harmonic vibrations acting simultaneously on a particle in perpendicular

directions. Here, the two vibrations are of the same time period but are of different

amplitudes and different phase angles.

From (6.3.1),

sinωt =y

b, (6.4)

∴ cosωt =

√

1−y2

b2. (6.5)

From Eq. (6.3.1),x

a= (sinωt cosα + cosωt sinα) . (6.6)

Substituting the value of sinωt and cosωt in Eq. (6.6)

x

a=

(

y

b· cosα+

√

1−y2

b2· sinα

)

, (6.7)

orx

a−y

b· cosα =

√

1−y2

b2· sinα. (6.8)

Squaring Eq. (6.8)

x2

a2+y2

b2· cos2 α−

2xy

ab· cosα =

(

1−y2

b2

)

sin2 α, (6.9)

orx2

a2+y2

b2(

sin2 α + cos2 α)

−2xy

ab· cosα = sin2 α, (6.10)

∴x2

a2+y2

b2−

2xy

ab· cosα = sin2 α. (6.11)

47

This represents the general equation of an ellipse. Thus, due to the superimposition of

two simple harmonic vibrations, the displacement of the particle will be along a curve

given by equation Eq. (6.11).

• Special cases

Figure 6.2: Special cases for Lissajous figures.

1. If α = 0 or 2π; sinα = 0 ; cosα = 1;

∴x2

a2+y2

b2−

2xy

ab= 0⇒

x

a−(y

b

)

= 0,

or

y =b

ax. (6.12)

This represents the equation of the straight line BD as in Fig 6.2 i.e., the particle

vibrates along the line BD.

2. If α = π; sinα = 0 ; cosα = −1;

∴x2

a2+y2

b2+

2xy

ab= 0⇒

(x

a+y

b

)

= 0,

y = −b

ax. (6.13)

This represents the equation of the straight line AC (Fig. 6.2).

48

3. If α = π/2 or 3π/2; sinα = 1 ; cosα = 0

x2

a2+y2

b2= 1. (6.14)

This represents the equation of the ellipse EHGF (Fig. 6.2) with a and b as the

semi-major and semi-minor axes.

For a = b Eq.(6.14) gives

x2 + y2 = a2, (6.15)

which is the equation of a circle of radius a Fig. 6.3.

Figure 6.3: For a = b Lissajous figure is a circle of radius a.

The phase relationship between transmitter and receiver signal is represented by a Lis-

sajous figure on the oscilloscope Fig. 6.4. If it is a straight line, the phase difference is 0

in the case of a positive slope and π in the case of a negative one.

6.3.2 Velocity of light in air

In order to measure the velocity of light in air (Fig. 6.4), the light path is extended by

∆l = 2 ·∆x, (6.16)

to produce a phase change of π: i.e. to travel this distance the light requires a time

∆t =1

2T =

1

2f, (6.17)

49

Figure 6.4: Diagram of the experimental set-up for measuring the velocity of light

in air.

where T is the period and f(= 50.1MHz) is the modulation frequency.

The velocity of light in air is thus expressed by

ca =∆l

∆t= 4f ·∆x. (6.18)

6.3.3 Velocity of light in medium

The velocity of light in water or synthetic resin, cm, is measured by comparing it with

the velocity of light in air ca (Fig. 6.5). In the first measurement (with the medium), the

light travels a distance l1 in time t1.

l1 = 2x1, (6.19)

t1 =laca

+lmcm

=l1 − lmca

+lmcm

, (6.20)

where la(= l1 − lm) is the distance travelled by the light in air and lm is the length of the

medium.

In the second measurement (no medium), the light travels a distance

l2 = l1 + 2∆x, (6.21)

in time

t2 =l1 + 2∆x

ca. (6.22)

The phase relationship between transmitter and receiver signals is the same in both cases,

so that

t2 = t1 + kT ; k = 0, 1, 2, · · · . (6.23)

50

Figure 6.5: Measuring the velocity of light in other media.

Taking k = 0 and equating Eqs. (6.20) and (6.22), we thus obtain the refractive index

n =cacm

=2∆x

lm+ 1. (6.24)

6.4 Equipment

Light velocity measuring apparatus, oscilloscope, synthetic resin block and water filled

tube.


The deviating mirror and the lenses are set up in such a way that the incident and

emergent light rays are parallel to the base plate (Fig. 6.1) and a maximum signal

(maximum intensity of light) reaches the receiving diode.

51

6.5.1 Adjustment of the apparatus

1. Place the two lenses in front of the transmitter and the receiver (the flat surface must

be towards the operating unit).

2. Placed the mirror as far to the operating unit as possible.

3. Adjust the position of the lens, which is in front of the transmitter, until a spot of

light is appear on the surface of the mirror (you can use a white paper in the path

of the light ray in order to see this spot of light).

4. The light beam will reflect to incident on the second part of the mirror, and the spot

of light will appear on its surface (use the white paper to see the spot).

5. Adjust the screws, fitted on the mirror, so that the beam of light is incident on the

convex surface of the lens, which is in front of the receiver.

6. Adjust the position of this lens until the light is concentrated on the receiving diode.

6.5.2 Measuring the velocity of light in air

1. The mirror is now placed as close to the operating unit as possible (zero point on the

scale) and a Lissajous figure appears on the oscilloscope.

2. Using the ’phase’ knob on the operating unit, transform the Lissajous figure into a

straight line.

3. The mirror is then slide along the graduated scale until the phase has changed by π,

i.e. until a straight line sloping in the opposite direction is obtained.

4. The mirror displacement ∆x is measured; the measurement should be repeated several

times (four times at least).

5. Record the results in Table 6.1.

6.Using Eq. (6.18) calculate the velocity of light in air.

52

Table 6.1: The results of a typical measurements for the determination of the velocity

of light in air.

f = 50.1 MHz

∆x (m)

ca (m/s)

average value: ca = cm/s

6.5.3 Measuring the velocity of light in medium

1. The medium (water-filled tube or the synthetic resin block) is placed in the path of

the ray so that its end faces are perpendicular to the optic axis.

2. Place the mirror directly behind the medium (part (a) of Fig. 6.5).

3. A straight line is obtained on the oscilloscope again with the ’phase’ knob.

4. The medium is then taken out of the path of the rays as shown in part (b) of Fig.

6.5.

5. Move the mirror until the Lissajous figure again shows the same phase difference; i.e,

until the same line is obtained again on the oscilloscope screen (part (c) of Fig. 6.5).

6. The mirror displacement ∆x is measured several times (four times at least).

7. Record the results in Table 6.2 for the synthetic resin and Table 6.3 for the water.

8. Using Eq. (6.24) calculate the refractive index of synthetic resin.

9. Using the measured values of ca and ns.r, calculate the velocity of light in synthetic

resin from equation cs.r = ca/ns.r.

10. Using Eq. (6.24) calculate the refractive index of water.

11. Using the measured values of ca and nH2O

, calculate the velocity of light in water

from equation cH2O

= ca/nH2O.

53


of light in synthetic resin.

length of synthetic resin block ls.r = cm

∆x (cm)

ns.r

average value: ns.r =

cs.r = ca/ns.r = cm/s


of light in water.

length of water tube lH2O = cm

∆x (cm)

nH2O

average value: nH2O

=

cH2O

= ca/nH2O= cm/s

54

7 Newton’s Rings

7.1 Objective

To determine the wavelengths for a given radius of curvature of the lens of Newton’s rings

apparatus.


In a Newton’s rings apparatus, monochromatic light interferes in the thin film of air

between the slightly convex lens and a plane glass plate. The wavelengths are determined

from the radii of the interference rings.

Figure 7.1: Experimental set-up for determining wavelength using the Newton’s

apparatus.

7.3 Theory

When a plano-convex lens of long focal length is placed on a plane glass plate, a thin film

of air is enclosed between the lower surface of the lens and the upper surface of the plate.

The thickness of the air film is very small at the point of contact and gradually increases

from the center outwards. The fringes produced with monochromatic light are circular.

55

The fringes are concentric circles, uniform in thickness and with the point of contact as

the center. When viewed with white light, the fringes are colored. With monochromatic

light, bright and dark circular fringes are produced in the air film.

7.3.1 Newton’s rings by transmitted light

Suppose the radius of curvature of the lens is R and the air film of thickness t is at a

distance of rn, from the point of contact with the glass plate.

Here interference is due to transmitted light (Fig. 7.2). Therefore, for the bright ring

Figure 7.2: Creating Newton’s rings.

2µt cos θ = nλ, (7.1)

where µ is the refractive index of the enclosed film and n = 0, 1, 2, · · · .

Here, θ is small, therefore cos θ = 1. For air, µ = 1:

2t = nλ ;n = 0, 1, 2, · · · . (7.2)

For the dark ring,

2µt cos θ = (2n− 1)λ

2, (7.3)

or

2t = (2n− 1)λ

2;n = 1, 2, 3, · · · . (7.4)

In Fig. 7.3,

r2n = t(2R− t). (7.5)

56

Figure 7.3: Geometry used to determine the thickness t.

But, 2R − t ≈2R since the thickness of the air film t is small with respect to the radius

of the lens.

r2n = 2Rt, (7.6)

or

t =r2n

2R. (7.7)

Substituting the value of t in Eq. (7.2) and Eq. (7.3.1) respectively:

r2n = nλR ;n = 0, 1, 2, · · · (for bright rings), (7.8)

r2n =

2n− 1

2λR ;n = 1, 2, , 3 · · · (for dark rings). (7.9)

When n = 0, for bright rings, r = 0.

Therefore, in the case of Newton’s rings due to transmitted light, the central ring is bright

(Fig. 7.4)

57

Figure 7.4: Newton’s rings due to transmitted light (the central ring is bright).

7.4 Equipment

Newton rings apparatus, Lens(f + 50mm), interference filters, translucent screen, Hg

vapour high-pressure lamp, power supply for Hg lamp, double condenser(f60mm), lens

holder, optical profile bench, plastic rule(20cm).


The Newton’s rings experiment is set up as shown in Fig. 7.1. The mercury vapour

high-pressure lamp with the double condenser (focal length 60mm) fitted, the lens holder

with the interference filter, the Newton’s rings apparatus, the lens holder with the lens of

focal length 50mm and a transparent screen about 40cm away from the lens are all set

up on the optical bench.

1. At the beginning of the experiment the path of the rays is adjusted, first without

colour filters, until interference rings can be observed on the screen.

2. The yellow filter is inserted in the lens holder and the room darkened.

58

3. By turning the three adjusting screws on the Newton’s rings apparatus to and fro, the

plano-convex lens is set on the plane-parallel glass plate so that the bright centre

of the interference rings is in the mid-point of the millimeter scale projected on the

screen.

• When making this adjustment, ensure that the lens and the glass plate only

just touch. This is achieved when no more rings emerge from the ring centre

when the adjusting screws are tightened up.

4. Measure the diameters Dn of the bright interference rings at the appropriate ordinals.

5. Record the measurements of Dn and the radii rn(=Dn

2) of the rings in table 7.1.

6. Repeat paragraphs 2, 4 and 5 for the blue colour filter at the appropriate ordinals.


Table 7.1: The results of a typical measurements for the determination of the wave-

length of light by using Newton’s rings apparatus.

colour

yellow

green

n 1 2 3 4 5 6 7 8

Dn (mm)

rn (mm)

r2n (mm2)

Dn (mm)

rn (mm)

r2n (mm2)

• Plot r2n as y-axis versus n as x-axis, then calculate the slope of the line obtained.

• Using Eq. (7.8) calculate the wave lengths of both yellow and blue lights.

λR = slope⇒ λ =slope

R; R = 12.141m

59

8 Specific charge of the electron - e/me

8.1 Objective

Determination of the specific charge of the electron (e/me) from the path of an electron

beam in crossed electric and magnetic fields of variable strength.


Electrons are accelerated in an electric field and enter a magnetic field at right angles

to the direction of motion. The specific charge of the electron is determined from the

accelerating voltage, the magnetic field strength and the radius of the electron orbit.

Figure 8.1: Experimental set-up for determining the specific charge of the electron.

8.3 Theory

8.3.1 Deflection of the electron in magnetic field

If an electron of mass me and charge e is accelerated by a potential difference V it attains

the kinetic energy:1

2mev

2 = eV, (8.1)

v2 =2eV

me

. (8.2)

where v is the velocity of the electron.

60

Figure 8.2: Circular motion in uniform magnetic field.

If the electron enters a uniform magnetic field of strength B acting perpendicular to the

direction of motion (Fig. 8.2), the Lorentz force acting on it is:

F = Bev. (8.3)

The direction of the force is perpendicular to both B and v.

If the field is uniform, the force is constant in magnitude and the beam then travels in a

circle of radius r. Since Bev is the centripetal force towards the center, we obtain

Bev =mev

2

r, (8.4)

v =rBe

me

. (8.5)

From Eq. (8.2) and Eq. (8.5)

2eV

me

=

(

rBe

me

)2

, (8.6)

ore

me

=2V

B2r2, (8.7)

from which (e/me) may be evaluated if V,B and r are known.

61

8.3.2 Magnetic field on the axis of a circular coil

To calculate the magnetic field B along the axis of a circular coil, the Biot-Savart law is

used.

Figure 8.3: Magnetic field at an axial point P for a circular coil.

Consider a circular coil of radius R located in the yz plane and carrying a steady current

I, as in Fig. 8.3. The magnetic field at an axial point P a distance x from the center of

the coil is

B =µ0IR

2

2(x2 +R2)3/2, (in the x direction) (8.8)

where µ0 is a constant, called permeability of free space:

µ0 = 4π × 10−7T.m/A

. When the coil has more than one turn, we can write

B =µ0NIR2

2(x2 +R2)3/2, (in the x direction) (8.9)

62

8.3.3 Helmholtz coils

The field along the axis of a single coil varies with the distance x from the coil. In order

to obtain a uniform field, Helmholtz used two coaxial parallel coils of equal radii(each of

radius R). In this case, when the same current flows round each coil in the same direction,

the resultant field B is uniform for some distance on either side of the point on their axis

midway between the coils. This may be seen roughly by adding the fields due to each coil

alone. Helmholtz coils were used in Thomson’s determination of e/m.

The magnitude of the resultant field B at the mid point can be found from our previous

formula for a coil of N turns(Eq. (8.9)). We now have x = R/2). Thus, for the two coils,

B = 2µ0NIR2

2(R2/4 +R2)3/2=

(

4

5

)3/2µ0NI

R. (8.10)

8.4 Equipment

Narrow beam tube, pair of Helmholtz coils, Power supply(0 − 600V dc), Power sup-

ply(universal), two digital multimeters, Connecting cords.

Figure 8.4: Wiring diagram for Helmholtz coils.


The experimental set up is as shown in Fig. 8.1. The electrical connection is shown in

the wiring diagram in Figures 8.4 and 8.5. The two coils are turned towards each other in

the Helmholtz arrangement. Since the current must be the same in both coils, connection

63

Figure 8.5: Wiring diagram for narrow beam tube.

in series is preferable to connection in parallel. The maximum permissible continuous

current of 3 A should not be exceeded. If the polarity of the magnetic field is correct, a

curved luminous trajectory is visible in the darkened room. If the trace has the form of a

helix this must be eliminated by rotating the narrow beam tube around its longitudinal

axis.

1. The power supply connected to the narrow beam tube and that connected to the

coils are adjusted so that a curved luminous trajectory is visible in the darkened

room.(The maximum permissible current of 3 A should not be exceeded)

2. Adjust the voltage connected to the beam tube to 100 V .

3. By means of varying the coil current I (the magnetic field), the radius of the orbit

can be adjusted so that it coincides with the radius defined by the 1st, 2nd, 3rd and

4th luminous traces (Fig. 8.6).

4. Record the values of I ′s in Table 8.1.

5. Repeat paragraphs 2 - 4 when the voltages are 150, 200 and 250 volts.

64

Figure 8.6: The luminous trajectories of the electron beam.


• Calculate the magnetic fieldB from Eq. (8.10) where R = 0.2m andN = 154 turns.

• Using Eq. (8.7) calculate the specific charge of the electron.

• Record the results of B and e/me in Table 8.1.

• Calculate the mean value of e/me.

65

Table 8.1: Current strength I (magnetic field B) and specific charge of the electron

e/me, in accordance with Eqs. (8.10) and (8.7) for various voltages V and various

radii r of the electron trajectories.

Radius

r = 0.02 m

r = 0.03 m

r = 0.04 m

r = 0.05 m

V (volts) 100 150 200 250

I (A)

B (T )

e/me (C/kg)

I (A)

B (T )

e/me (C/kg)

I (A)

B (T )

e/me (C/kg)

I (A)

B (T )

e/me (C/kg)

Mean value of e/me = C/kg

66

9 Diffraction grating

9.1 Objective

1.To determine the wavelengths of the mercury spectral lines using plane transmission

grating.


The wave lengths of the spectral lines of a mercury vapour lamp is determined with a

grating spectroscope.

9.3 Theory

Analytical treatment of interference of light due to double slit shows that the intensity at

a point on the screen is given by

I = 4a2 cos2δ

2, (9.1)

where a is the amplitude of the interfered waves and δ is the phase difference.

• When the phase difference δ = n(2π), or the path difference Γ = nλ, where n =

0, 1, 2, 3 · · · , then

I = 4a2. (9.2)

Thus, intensity is maximum when the path difference is a whole number multiple

of wave length.

• When the phase difference δ = (2n + 1)π, or the path difference Γ = (2n + 1) λ2,

where n = 0, 1, 2, 3 · · · , then

I = 0. (9.3)

Thus, intensity is minimum when the path difference is a whole number multiple of

wave length.

9.4 Fraunhofer diffraction at double slit

In Fig. 9.1, AB and CD are two rectangular slit parallel to one another and perpendicular

to the plane of the paper. the width of each slit is a and the width of the opaque portion is

67

b. L is a collecting lens and MN is a screen perpendicular to the plane of the paper. P is

a point on the screen such that OP is perpendicular to the screen. Let a plane wavefront

be incident on the surface of XY . All the secondary waves travelling in a direction parallel

to OP come to focus at P . Therefore, P corresponds to the position of the central bright

maximum.

In this case, the diffraction pattern has to be considered in two parts:

Figure 9.1: Fraunhofer diffraction at double slit.

1. The interference phenomenon due to the secondary waves emanating from the corre-

sponding points of the two slits.

2. The diffraction pattern due to the secondary waves from the two slits individually.

For calculating the positions of interference maxima and minima the diffracting angle is

denoted as θ and for the diffraction maxima and minima it is denoted as φ. Both the

angles θ and φ refer to the angle between the direction of the secondary waves and the

initial direction of the incident light.

9.4.1 Interference maxima and minima

Consider the secondary waves traveling in a direction inclined at an angle θ with the

initial direction.

68

Figure 9.2: Fraunhofer diffraction at double slit.

In the 4ACN (Fig. 9.2)

sin θ =CN

AC=

CN

a+ b, (9.4)

or

CN = (a+ b) sin θ. (9.5)

If this path difference is equal to odd multiples of λ2, θ gives the direction of minima due

to interference of the secondary waves from the two slits.

∴ CN = (a+ b) sin θn = (2n+ 1)λ

2. (9.6)

Putting n = 1, 2, 3 · · · , the values of θ1, θ2, θ3 · · · , corresponding to the directions of

minima can be obtained from the equation

sin θn =(2n+ 1)λ

2(a+ b). (9.7)

On the other hand, if the secondary waves travel in a direction θ′

such that the path

difference is even multiples of λ2, then θ

′

gives the direction of the maxima duo to the

interference of light waves emanating from the two slits.

∴ CN = (a+ b) sin θ′

n = 2nλ

2. (9.8)

69

or

sin θ′

n =nλ

a+ b. (9.9)

Putting n = 1, 2, 3 · · · , the values of θ1, θ2, θ3 · · · , corresponding to the directions of

maxima can be obtained.

From Eq. (9.7)

sin θ1 =3λ

2(a+ b)(9.10)

and

sin θ2 =5λ

2(a+ b), (9.11)

∴ sin θ2 − sin θ1 =λ

a+ b. (9.12)

Thus the angular separation between any two consecutive minima (or maxima) is equal to

λa+b

. The angular separation is inversely proportional to d (= a+ b), the distance between

the two slits.

9.4.2 Diffraction maxima and minima

Consider the secondary waves traveling in a direction inclined at an angle φ with the

initial direction of the initial lights.

If the path BM is equal to λ the wave length of light used, then φ will give the direction

of the diffraction minimum (Fig. 9.2). That is, the path difference between the secondary

waves emanating from the extremities of a slit (i.e., points A and B) is equal to λ.

Considering the wavefront on AB to be made up of two halves, the path difference between

the corresponding points of the upper and lower half is equal to λ2. The effect at P

′

due

to the wavefront incident on AB is zero. Similarly for the same direction of the secondary

waves, the effect at P′

due to the wavefront incident on the slit CD is also zero. In

general,

a sinφn = nλ. (9.13)

Putting n = 1, 2, 3 · · · , the values of φ1, φ2, φ3 · · · , corresponding to the directions of

diffraction minima can be obtained.

9.4.3 Plane transmission diffraction grating

A diffraction grating is an extremely useful device and in one of its form it consists of a

very large number of narrow slits side by side. The slits are separated by opaque spaces.

70

The distance between two slits (a+ b or d) is known as the grating element and it is equal

to the reciprocal of the number of lines per centimeter (N):

d =1

N. (9.14)

When a wavefront is incident on a grating surface, light is transmitted through the slits

and obstructed by the opaque portions. Such a grating is called a transmission grating.

The secondary waves from the positions of the slits interfere with one another, similar to

the interference of waves in Young’s experiment.

9.4.4 Determination of the wave length of a spectral line using plane trans-

mission grating

In the laboratory, the grating spectrum of a given source of light is obtained by using a

spectrometer. Initially all the adjustments of the spectrometer are made and it is adjusted

for parallel rays.

If the source of light emits radiations of different wavelengths, then the beam gets dis-

persed by the grating and in each order a spectrum of the constituent wavelength is

observed. To find the wavelength of any spectral line, the diffracting angles are noted in

the first and the second orders and using the equation

d sin θn = nλ, (9.15)

the wavelength of the spectral line can be calculated.

9.5 Equipment

Spectrometer, lamp holder, spectral lamp Hg, power supply for spectral lamps, diffraction

grating.


The experiment is set up as shown in Fig. 9.3. The spectrometer and the grating should

be adjusted in accordance with the procedure (steps 1-3).

1. Illuminate the slit of the collimator by the mercury light.

71

Figure 9.3: Experimental set-up for determining dispersion in diffraction grating.

2. Adjust the position of the telescope such that the image of the slit is obtained at

the position of the vertical cross-wire in the field of view of the telescope. In this

position the reading on the circular scale must be zero.

3. Place the grating on the table of the spectrometer such that its surface is normal to

the incident beam. We can adjust the grating, normally to the incident beam, by

looking to the spectrum through the telescope and then rotating the table until the

spectrum reverse its direction. In this position the grating surface is normal to the

incident beam.

4. Measure the angles of diffraction θ1L, θ1R, θ2L and θ2R corresponding to the first and

second orders for each spectral line, and record the measurements in Table 9.1, (θL

and θR denote the angles measured from the left and from the right sides of the

central image as in Fig 9.4).

5. Calculate the angles of diffraction θ1 and θ2, where

θ1 =θ1L + θ1R

2,

72

Figure 9.4: Dispersion in diffraction grating.

and

θ2 =θ2L + θ2R

2.

6. Record the values of θ1 and θ2 in Table 9.1.


(a) Calculate the wavelengths λ1 and λ2 for the first and second orders, for each spectral

line from the relation

d sin θn = nλ, d =1

N.

(b) Find the average value of the wavelength for each line:

λ =λ1 + λ2

2.

(b) Record the results in Table 9.1.

73

Table 9.1: Typical measurements for determining the wavelengths of the mercury

spectral lines using plane transmission grating.

1st order

2nd order


θ1L

θ1R

θ1

λ1 (nm)

θ2L

θ2R

θ2

λ2 (nm)

λ = λ1+λ2

2(nm)

74

10 Laws of lenses

10.1 Objective

1. To determine the focal length of an unknown convex lens by measuring the distances

of image and object (using thin lens equation method).

2. To determine the focal length of a convex lens using Bessel’s method.


The focal lengths of unknown lenses are determined by measuring the distances of image

and object and by Bessel’s method.

10.3 Theory

The focal length of a lens can be determined by several techniques. Some of these are less

difficult to use than others and some are more accurate. The following two subsections

are a brief description of some of the techniques.

10.3.1 Thin Lens Equation

The relationship between the focal length f of a lens, the object distance u and the image

distance v is obtained from geometrical optics. Three particular rays, the focal ray, the

parallel ray and the central ray, are used to construct the image (Fig. 10.1). From the

laws of similar triangles,I

O=

v

u(10.1)

andO

I=

f

v − f(10.2)

where I is the image size and O is the object size. From Eq. (10.3.1) and Eq. (10.3.2),

we can writeu

v=

f

v − f⇒ uv − uf = vf ⇒ uv = (u+ v)f, (10.3)

from which we obtain the lens formula

1

f=

1

u+

1

v, (10.4)

75

Figure 10.1: Image construction with three principal rays.

10.3.2 Bessel’s method

Bessel developed a method of conjugate foci which is more accurate than the above

method. The distance between object and image screen, L, is made longer than four

times the focal length. If the lens is moved back and forth along the axis, the image will

be in focus at two positions of the lens; in one position the image is magnified, in the

other it is reduced. The focal length is calculated from the equation

f =L2 − d2

4L. (10.5)

where d is the distance between the two lens positions that produce an image for the given

conjugate plane separation as illustrated in Fig 10.2. The focal length of the convex lens

can therefore be determined from the measured values of L and d.

• Derivation of Eq. (??)

The distance between the object and the image (Fig. 10.2) is given by

L = u1 + v1. (10.6)

The displacement between the first and the second positions of the lens is

d = v1 − v2. (10.7)

Since u1 = v2 (the object distance in case 1 = image distance in case 2),

∴ d = v1 − u1. (10.8)

76

Figure 10.2: Determination of focal length by using Bessel’s method.

If we solve the equations (10.6) and (10.8) for u1 and v1 we obtain

v1 =1

2(L+ d), (10.9)

and

u1 =1

2(L− d). (10.10)

Substituting into the lens formula gives

f =L2 − d2

4L. (10.11)

The focal length of the convex lens can therefore be determined from the measured values

of L and d.

10.4 Equipment

Lenses, translucent screen, screen with arrow slit, double condenser f = 60 mm, optical

profile-bench, power supply 0-12V DC.

77


The experiment is set up as shown in Fig. 10.3. A parallel light beam is produced with

the lamp and the double condenser.

Figure 10.3: Experimental set-up for determining the focal length of a lens.

10.5.1 Determination of the focal length of an unknown convex lens by using

thin lens equation method

1. A parallel light beam is produced with the lamp and the double condenser.

2. Put the object (screen with arrow slit) directly behind the condenser, and a clear

image is projected on to the screen with a lens.

3. The distances of image and object from the lens are measured (assume that the lens

is thin).

4. The measurement of distances of image and object is repeated, with the lens and the

screen in different positions.

5. Record the measurements in Table 10.1.

78

10.5.2 Determination of the focal length of a convex lens by using Bessel’s

method.

1. The distance between the object and the screen, L, is made longer than four times

the focal length of the lens.

2. Adjust the position of the lens until you obtain a clear image on the screen, and

record the scale reading of the optical profile-bench (the image is magnified).

3. Alter the position of the lens so that the image and object distances are transposed,

and record the new reading of the scale (the image is reduced).

4. Find the distance d between the initial and the final positions of the lens, and record

the value of d in Table 10.2.

5. Repeat steps (2-4) four times and record the value of d in Table 10.2.

10.6 Measurements and Calculations

Part I

Table 10.1: The results of a typical measurements for the determination of the focal

length of a convex lens by using thin lens equation method.

u(cm) 15 17 19 21 23

v(cm)

f(cm)

• Compute the focal length of the lens from Eq. (10.4).

• Find the average value of the focal length.

f̄ = cm.

79

Part II

Table 10.2: The results of a typical measurements for the determination of the focal

length of a convex lens by using Bessel’s method.

Trial 1 2 3 4

d(cm)

• Calculate the average value of the distance d.

• Compute the focal length of the lens from Eq. (10.5).

L = cm

d̄ = cm

f = cm

80

Documents

LaboratoryExperimentsinOptics - site.iugaza.edu.pssite.iugaza.edu.ps/helegla/files/2010/02/optics.lab.tex.pdf · IslamicUniversity-Gaza FacultyofScience DepartmentofPhysics LaboratoryExperimentsinOptics