In the figure shown, a screen is placed normal to the line joining the two point coherent sources `S_1 and S_2`. The interference pattern consists of concentric circles.

(a)Find the radius of the nth bright ring.
(b) If d=0.5 mm, `lambda=5000 Å` and D=100 cm, find the radius of the closest second bright ring.
(c) Also, find the value of n for this ring.

Options

A. `D sqrt(1(1 - (n lambda)/(d)))`
B. `D sqrt(2(1 - (n lambda)/(d)))`
C. `2D sqrt(2(1 - (n lambda)/(d)))`
D. `D sqrt(2(1 - (n lambda)/(2d)))`

Answer

b

Explanation

The optical path difference at P is
`Delta x = S_(1) P - S_(2) P = cos theta`
`:' cos theta = 1 - (theta^(2))/(2)` for small `theta`
`:. Delta x = d(1 - (theta^(2))/(2))`
`= d[1 - (y^(2))/(2D^(2))]`, where `D + d = D`
For nth maxima,
`implies Delta x = n lambda`
`d [1 - (y^(2))/(2D^(2))] = n lambda`
y = radius of the nth brigth ring
`= D sqrt(2(1 - (n lambda)/(d)))`
At the central maxima, `theta = 0`
`Delta x = d = n lambda`
`implies n = (d)/(lambda) = (0.5)/(0.5 xx 10^(-3)) = 1000`
Hence, for the closest second brigth ring, `n = 998`.
Putting values, `r = 6.32cm`.