An interference is observed due to two coherent sources `S_1` placed at origin and `S_2` placed at `(0, 3lambda, 0)`. Here, lambda is the wavelength of the sources. A detector D is moved along the positive x-axis. Find x-coordinates on the x-axis (excluding `x = 0 and x= oo`) where maximum intensity is observed.

Options

A. `x = 4 lambda`
B. `x = 7 lambda//4`
C. `x = 5 lambda//4`
D. `x = 3 lambda`

Answer

a.,c

Explanation

At ` x = 0`, path difference is `3 lambda`. Hence,third-order maximum will be obtained. At `x = oo`, path difference is zero. Hence, zero-order maxima is obtained. In between, first - and second-order maxima will be obtained. First order maxima:
`S_(2) P - S_(1) P = lambda` or `sqrt(x^(2) + 9lambda^(2)) - x = lambda`
or `sqrt(x^(2) + 9 lambda^(2)) = x + lambda`
Squaring this, we get `x^(2) + 9 lambda^(2) = x^(2) + lambda^(2) + 2 x lambda`
Solving this, we get `x = 4 lambda`
Second-order maxima:
`S_(2) P - S_(1) P = 2 lambda` or `sqrt(x^(2) + 9 lambda^(2)) - x = 2 lambda`
or ` sqrt( x^(2) + 9 lambda^(2)) = (x + 2 lambda)`
Squaring both sides, we get
`x^(2) + 9 lambda^(2) = x^(2) + 4 lambda^(2) + 4 x lambda`
Solving this, we get `x = (5)/(4) lambda = 1.25 lambda`
Hence, the desired x-corrdinates are
`x = 1.25 lambda` and `x = 4 lambda`