Question

Derive a formula for the radius of of the dark ring in a Newton’s Ring.

Answer 1

Answer:

Let R be the radius of curvature of lens surface. Let nth dark ring is obtained at point B. Let rn is the radius of nth dark ring.

In triangle OCB

R2 = (R−t)2 + r2n

R2 = R2 - 2Rt + t2 + r2n

⸫ r2n = 2Rt …………………………………..(1)

As t is very small higher orders of t can be neglected.

For dark ring 2μtcos(r+θ) =nλ

cos(r+θ) =1 for normal incidence and large radius of curvature

⸫ t = nλ / 2μ

Substituting in equation (1)

⸫ r−n2 = Rnλ / μ

⸫ rn = n−−√

For air film μ=1,

The experimental set up is as shown in the figure below.

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The light from the source is rendered parallel by means of a converging lens. The parallel beam of light is intercepted by means of a glass plate inclined at an angle of 45° so as to get the normal incidence. This light illuminates the air film and the interference pattern is observed through a traveling microscope focussed on it.

The interference pattern consists of dark and bright concentric rings with dark center when observed in reflected light. The cross wire of the eyepiece of an traveling microscope is focussed on nth dark ring and microscopic readings(main scale reading and Vernier scale readings) are noted when the film is air.

D2n = 4nλR-------------------------------(1)

The traveling microscope is moved and focussed on (n+p)th dark ring,and again microscopic readings are noted when the film is air.

D2(n+p) = 4(n+p)λR -------------------------------(2)

From equation (1) and (2),, D2(n+p) - D2n = 4pλR …………………………………(3)

Now, pour the liquid whose refractive index is to be determined. Now air film is replaced by liquid film .

Repeat the same procedure and find diameter of nth and (n+p)th ring by taking main scale and Vernier scale reading when the film is liquid.

D′2(n+p) - D′2n = 4pλR / μ………………………………….(4)

Divide equation (3) and (4)

⸫ μ =(D2(n+p) - D2 )air / (D′2(n+p)- D′2 )liquid