(a) In Illustration 2.20, how many dark rings will be observed on the wall? (b) What is the Path difference at point P?

Explanation

(a) Path difference corresponding to minima in our given range is `lambda//2, 3 lambda//2, 5 lambda//2`.
Hence, three dark rings will be observed on the wall.
(b) Path difference between the waves reaching at P,
from Fig. 2.20 `Delta x = S_(2) A`
`S_(2) P - S_(1) P = Delta x or d cos theta = Delta x`
`cos theta = (Delta x)/(d)`
But `tan theta = (y)/(D)`
`implies cos theta = (sqrt(D ^(2) - Y^(2)))/(D)`
`(D sqrt(1 - (y^(2))/(D^(2))))/(D)`
`= (1 - (y^(2))/(D^(2)))^(1//2) = (1 - (Y^(2))/(2D^(2)))`
From (i) `Delta x = d cos theta = d (1-(y^(2))/(2D^(2)))`
From (i) and (ii), we can find the radius of any dark of bright ring.