Two identical sources each of intensity `I_(0)` have a separation `d = lambda // 8`, where `lambda` is the wavelength of the waves emitted by either source. The phase difference of the sources is `pi // 4` The intensity distribution `I(theta)` in the radiation field as a function of `theta` Which specifies the direction from the sources to the distant observation point P is given by

Options

A. `I(theta) = I_(0)cos^(2)theta`
B. `I(theta)=(I)_(0)/(4)cos^(2)((pi theta)/(8))`
C. `I(theta)=4I_(0)cos^(2)((pi)/(8)(sin theta+1))`
D. `I(theta)= I_(0) sin^(2)theta`

Answer

C

Explanation

`Delta x = d sin theta = (lambda)/(8) sin theta`
`Delta phi = (2pi)/(lambda)xx(lambda)/(8) sin theta + Delta phi_(0)` (`Delta phi_(0)` is initial phase difference between the slits)
`Delta phi = (pi)/(4) sin theta + (pi)/(4)`
`Delta phi = (pi)/(4) (1+sin theta)`
Resultant intensity `I = 4I_(0) cos^(2) ((Delta phi)/(2))`
`I = 4I_(0) cos^(2) {(pi)/(8) (1+sin theta)}`