(a) There are two sets of apparatus of Young’s double slit experiment. In set A, the phase difference between the two waves emanating from the slits does not change with time, whereas in set B, the phase difference between the two waves from the slits changes rapidly with time. What difference will be observed in the pattern obtained on the screen in the two set ups?

(b) Deduce the expression for the resultant intensity in both the above-mentioned set ups (A and B), assuming that the waves emanating from the two slits have the same amplitude A and same wavelength λ.

OR

(a)The two polaroids, in a given set up, are kept ‘crossed’ with respect to each other. A third polaroid, now put in between these two polaroids, can be rotated. Find an expression for the dependence of the intensity of light I, transmitted by the system, on the angle between the pass axis of first and the third polaroid. Draw a graph showing the dependence of I on ϴ.

(b)When an unpolarized light is incident on a plane glass surface, find the expression for the angle of incidence so that the reflected and refracted light rays are perpendicular to each other. What is the state of polarization, of reflected and refracted light, under this condition?

a) In Set A, as the phase difference between the two waves emanating from the slits does not change with time so alternate bright and dark bands would be seen on the screen as per Young’s double slit experiment as two slits are acting as coherent sources whereas in Set B, the Phase difference changes rapidly with time, the light waves coming out from two independent sources of light will not have any fixed phase relationship and would be incoherent, when this happens, the intensities on the screen will add up and only bright light will be seen.

b) Given: -

The waves emanating from the two slits have the same amplitude A and same wavelength λ

Derivation: -

Let the displacement produced by wave emanating from slit 1 be y₁ and from slit 2 be y₂

So, we can write,

y₁=Acosωt

And,

y₂=Acos(ωt+ϕ)

Where is the phase difference between the two sources, and A is the amplitude,

the resultant displacement will be given by,

y = y₁ + y₂

so, after substituting the values we get,

using the trigonometric identity,

We get,

So, the amplitude of the resultant displacement is,

As we know that the intensity is directly proportional to the square of the amplitude, so,

I₀∝A²

And

I∝A^'2

So, we can write,

Which gives,

Conclusions: -

The resultant intensity is given as,

For set A,

As is constant, the value of I remains constant for a particular band on the screen, which leads to formation of alternate bright and dark bands.

For set B,

As is changing rapidly with time, the value of I changes rapidly with time on the screen. The value of I changes so rapidly from 0 to 4I₀ and again back to zero so rapidly, so it seems to the human eye that the intensity is constant as I = 4I₀ and it seems that constructive inference is occurring.

OR

a) Given: -

The intensity of the incident light I₀

Derivation: -

After passing the first polaroid (P₁) the intensity be I₁

We know that I₁ reduces to half

After passing the third polaroid (P₃) the intensity be I₂

We can write by malus’ law as,

Also the angle between the pass axis of P₃ and P₂ is 90 - ϴ,

So the intensity of light transmitted through P₂ is,

a) When unpolarized light is incident on a plane glass surface, the reflected light is polarized with its electric vector perpendicular to the plane of incidence when the refracted and reflected rays make a right angle with each other. Thus, when reflected wave is perpendicular to the refracted wave, the reflected wave is a totally polarized wave. The angle of incidence in this case is called Brewster’s angle and is denoted by i_B. We can see that i_B is related to the refractive index of the denser medium.

Since we have,

from Snell’s law we get,

So, finally,

μ=tani_B

The above expression is called the Brewster’s Law.

The state of the above light is partially polarized because When such light is viewed through a rotating analyzer, one sees a maximum and a minimum of intensity but not complete darkness.