Answers of Chapter Speed of light Grade 12

 1.

Measuring the velocity of light has great importance in physics. Some of them are:-

(a) By the determination of velocity of light in vacuum and in a medium, we can determine the refractive index of the medium.

(b) By the determination of velocity of light, we can estimate the energy equivalence of a mass from Einstein's mass energy relation.

(c) The relativistic mass of a particle of rest mass 'M0 'can be determined with the help of the relation:

 M=M01−v2c2√$\frac{{\mathrm{M}}_0}{\sqrt{1-\frac{{\mathrm{v}}^2}{{\mathrm{c}}^2}}}$

by knowing the speed of particle v and speed of light is vacuum c.        

(d) we can know the variation of speed of light in different media.

 

 

2.

 

3.

 

4.

The advantages of Foucault’s method are:

a. It can be performed in laboratory as it covers the small area.

 b. Speed of the light in any optical medium can be determined.

c. It justified the validity of wave theory of light as velocity of light in water found to be less than velocity of light in air.

 The disadvantages of Foucault’s method are:

a. The image obtained is very faint due to reflection and refraction of light at various surfaces thus makes the observation difficult.

 b. Due to small displacement of the image accurate measurement cannot be obtained.

                                                                                                                                                          

 

5.

 

6.

Foucault’s method is used to measure the speed of light in laboratory.

7.

Foucault’s method is used to measure the speed of light in water.

8.

The advantages of Foucault’s method over Fizeau’s method of measuring speed of light are as follows;

a. Foucault’s method can be done in laboratory but Fizeau’s method cannot be used in laboratory since it requires very large distance in the range of 10-12 km.

b. Foucault’s method can be used to measure velocity of light in various media including gases but Fizeau’s method can be used to measure velocity of light in air of vacuum only.

 

Long type question

1.           

Light from a source S is focused by a lens L1 on to the edge of a toothed wheel W which has N teeth. It passes through the gaps between the teeth and on to a second lens L2, which produces a beam of parallel light. This travels across a known distance d to a third lens L3 and on to a concave mirror M. It is then reflected back and when it meets plate P it is reflected into the eyepiece E (Figure 1).

The wheel is now rotated until a speed is reached at which the pulse of light leaving through one gap returns to the wheel when the next tooth has taken the place of the gap (Figure 2). No light will therefore be observed reaching the eyepiece. If the wheel is rotating at n revolutions per second then

Distance travelled by light = 2d
Transit time = 1/2nN
Speed of light (c) = 4nNd

Fizeau's values were:
2d = 17.26 km, N = 720, n = 12.6 revs per second, giving a value of 3.13 x 108 ms-1 for the speed of light.

The light must be monochromatic to give a parallel beam and the teeth on the wheel should be blackened to stop unwanted reflections.

 

2.

Theory: Light from a source S is focused by a lens L1 on to the edge of a toothed wheel W which has N teeth. It passes through the gaps between the teeth and on to a second lens L2, which produces a beam of parallel light. This travels across a known distance d to a third lens L3 and on to a concave mirror M. It is then reflected back and when it meets plate P it is reflected into the eyepiece E

 

he wheel is now rotated until a speed is reached at which the pulse of light leaving through one gap returns to the wheel when the next tooth has taken the place of the gap (Figure 2). No light will therefore be observed reaching the eyepiece. If the wheel is rotating at n revolutions per second then

Distance travelled by light = 2d
Transit time = 1/2nN
Speed of light (c) = 4nNd

Fizeau's values were:
2d = 17.26 km, N = 720, n = 12.6 revs per second, giving a value of 3.13 x 108 ms-1 for the speed of light.

The light must be monochromatic to give a parallel beam and the teeth on the wheel should be blackened to stop unwanted reflections.

 

 

3.                   

The experimental arrangement of Foucault’s method are given below

Experimental arrangement:

 

 

 

The rays of light from a bright source S are allowed to fall on convex lens L, which will bring them to focus at point I in the absence of plane mirror XY, which is capable of rotation about an axis through the point Q. The plane mirror make the ray to meet at point P, the pole of the concave mirror M such that PQ=IQ=d say. When the plane mirror is stationary, the rays of the light after reflection from concave mirror retrace their path and finally image coincident with S if the glass plate is placed at 450to the optical axis of the lens, then the returning light is reflected from it so as to produce the image I’( instead the image coincident with S)

Theory

When the plane mirror is rotated about its axis though the point Q, the intermittent image of the source is seen through the eyepiece,it is because the light falls on the concave mirror for a small fraction of revolution. As the speed of revolution is increased slowly, a stage comes, when image is seen continuously due to persistence of vision. It happens when mirror is rotated at the speed of more than 10r.p.s. as the distance between the plane mirror and the concave mirror is very small, negligible is compared to the velocity of light therefore , the light returning to the plane mirror ( after the reflection from the concave mirror)will find it practically in the same position . As a result, when plane mirror is rotating at low speed, the image will be seen still at I’

Now, suppose the speed of rotation of the plane mirror is increased. The light reflected from the plane mirror in the position XY, on returning from the concave mirror will find it in position X’Y’ i.e. displaced through, say angle ɵ the reflected ray will turn through angle 2ɵ. To the eye, the rays will appear to diverge from I and the image will be seen to shift to position I”. The displacement II’’ can be measure with the micrometer attached to the eyepiece.

Let C be the velocity of the light d be the distance between the plane mirror and concave mirror and n be the no. of revolutions made per second by the rotating mirror.

The time taken by the light to cover distance 2d i.e. from Q to P and back to Q is given by

T=2d/c…………………….1

As the plane mirror make n rotation per second. It covers an angle 2∏n in one second. Therefore, time taken by the mirror to rotate thought angle ɵ is given by

T=θ2πn$\frac{\theta }{2\pi \mathrm{n}}$………………2

From 1 and 2 we get

C=4πndθ$\frac{4\pi \mathrm{n}\mathrm{d}}{\theta }$…………………3

To find ɵ                                      

Let a and b be the distance of the plane mirror and the source of the light from the optical centre of the convex lens.

Now angle between two reflected rays QI and QI’ is 2ɵ therefore

2ɵ=II’/d

Or, II’=2ɵd……………….4

The image S and S’ formed by the lens are the images of I and I’

From the relation

Size of image/size of object=distance of image/ distance of object

Then we have,SS’/II’=OS/OI…………5                              

Where OS=b and OI=a+d

Now SS’=I’I”

Let the displacement I’I” in the image be x then, SS’=x

Then from equation 5 we have

x/II’=b/a+d

or, II’=(a+d)xb$\frac{\left(a+d\right)x}{b}$…………………………6

From 4 and6 we have

ɵ= (a+d)x2bd$\frac{\left(a+d\right)x}{2bd}$………………7

on putting the value of ɵ in equation in 3 we get

C=8πnbd2(a+d)x$8\frac{\pi nb{d}^2}{\left(a+d\right)x}$

This equation gives the speed of light in term of speed rotation. According to the Newton’s corpuscular, the velocity of the light in water should be greater than air. By using the Foucault method, It was found that s’/s is greater i.e. the velocity of light in water is smaller than velocity of light in air. Also the ratio of c/c’ c’ being speed of light in water, equal to the refractive index supporting the wave theory of light.

 

 

4.

The experimental arrangement of Foucault’s method are given below

Experimental arrangement:

 

 

The rays of light from a bright source S are allowed to fall on convex lens L, which will bring them to focus at point I in the absence of plane mirror XY, which is capable of rotation about an axis through the point Q. The plane mirror make the ray to meet at point P, the pole of the concave mirror M such that PQ=IQ=d say. When the plane mirror is stationary, the rays of the light after reflection from concave mirror retrace their path and finally image coincident with S if the glass plate is placed at 450to the optical axis of the lens, then the returning light is reflected from it so as to produce the image I’( instead the image coincident with S)

Theory

When the plane mirror is rotated about its axis though the point Q, the intermittent image of the source is seen through the eyepiece, it is because the light falls on the concave mirror for a small fraction of revolution. As the speed of revolution is increased slowly, a stage comes, when image is seen continuously due to persistence of vision. It happens when mirror is rotated at the speed of more than 10r.p.s. as the distance between the plane mirror and the concave mirror is very small, negligible is compared to the velocity of light therefore , the light returning to the plane mirror ( after the reflection from the concave mirror)will find it practically in the same position . As a result, when plane mirror is rotating at low speed, the image will be seen still at I’

Now, suppose the speed of rotation of the plane mirror is increased. The light reflected from the plane mirror in the position XY, on returning from the concave mirror will find it in position X’Y’ i.e. displaced through, say angle ɵ the reflected ray will turn through angle 2ɵ. To the eye, the rays will appear to diverge from I and the image will be seen to shift to position I”. The displacement II’’ can be measure with the micrometer attached to the eyepiece.

Let C be the velocity of the light d be the distance between the plane mirror and concave mirror and n be the no. of revolutions made per second by the rotating mirror.

The time taken by the light to cover distance 2d i.e. from Q to P and back to Q is given by

T=2dc$\frac{2d}{c}$…………………….1

As the plane mirror make n rotation per second. It covers an angle 2∏n in one second. Therefore, time taken by the mirror to rotate thought angle ɵ is given by

T=ɵ/2∏n………………2

From 1 and 2 we get

C=4∏nd/ɵ…………………3

To find ɵ                                      

Let a and b be the distance of the plane mirror and the source of the light from the optical centre of the convex lens.

Now angle between two reflected rays QI and QI’ is 2ɵ therefore

2ɵ=II’/d

Or, II’=2ɵd……………….4

The image S and S’ formed by the lens are the images of I and I’

From the relation

Size of image/size of object=distance of image/ distance of object

Then we have,SS’/II’=OS/OI…………5                              

Where OS=b and OI=a+d

Now SS’=I’I”

Let the displacement I’I” in the image be x then, SS’=x

Then from equation 5 we have

x/II’=b/a+d

or, II’=(a+d)xb$\frac{\left(a+d\right)x}{b}$…………………………6

From 4 and6 we have

ɵ= (a+d)x2bd$\frac{\left(a+d\right)x}{2bd}$………………7

on putting the value of ɵ in equation in 3 we get

C=8πnbd2(a+d)x$\frac{8\pi nb{d}^2}{\left(a+d\right)x}$

This equation gives the speed of light in term of speed rotation.

The advantages of Foucault’s method over Fizeau’s method of measuring speed of light are as follows;

a. Foucault’s method can be done in laboratory but Fizeau’s method cannot be used in laboratory since it requires very large distance in the range of 10-12 km.

b. Foucault’s method can be used to measure velocity of light in various media including gases but Fizeau’s method can be used to measure velocity of light in air of vacuum only.

 

5.

The experimental arrangements are:

 

The experimental arrangement of Michelson method is as shown in figure. It consists of three mirrors such as octagonal mirror (M1), concave mirror (M2) & plane mirror (M3). Light from the source (S) incident at an angle of 450 on one face of octagonal mirror (M1) then, the reflected light from this face falls on a distant concave mirror (M2).

With the help of plane mirror (M3)placed at a center of curvature of concave mirror (M2), the light is returned back and falls on the face of mirror (M1) again at angle of 45°. The light reflected from this face is collected by telescope and observed by eye. If light returning from the mirror (M2) will not in general incident at an angle 45°lights is not observed by telescope.

The rotation of mirror (M1) is so adjusted that, the face 1 of mirror occupies exactly the same position as was occupied by face 2 during the time light travels from (M1) to (M2) and returning back to (M1). Then image of source will be reappearing.

 

If‘d’ be the distance between mirror (M1) and (M2 ) and ‘c’ be the velocity of light, then time taken by light to travel from (M1 ) to (M2 ) and returning back to (M1 ) is

Given by, t=2d/c

if ‘m’ be the no. of faces of polygonal mirror and ‘n’ be the no. of revolution per second. Then angle rotated by mirror during the time t i.e. ɵ=2π/m

and time taken by mirror to rotate by angle θ i.e.

t=θ2πn$\frac{\theta }{2\pi \mathrm{n}}$

2dc$\frac{2\mathrm{d}}{\mathrm{c}}$ =θ2πn$\frac{\theta }{2\pi \mathrm{n}}$

c=4πndθ$\frac{4\pi \mathrm{n}\mathrm{d}}{\theta }$

Since, ɵ=2π/m

Then, ɵ=2dnm.