A Young's double slit interference arrangement with slits `S_(1)` and `S_(2)` is immersed in water (refractive index `=4//3`) as shown in the figure. The positions of maxima on the surface of water are given by `x^(2) = p^(2)m^(2)lambda^(2)-d^(2)`, where `lambda` is the wavelength of light in air (reflactive index = 1), 2d is the separation between the slits and m is an integer. The value of P is ..........

Options

A. `1`
B. `2`
C. `3`
D. `4`

Answer

C

Explanation

Optical path length,
`S_(2)P = rsqrt(x^(2) + d^(2)) = (4)/(3)sqrt(x^(2) + d^(2))`
Optical path length `S_(2)P = sqrt(x^(2) + d^(2))`
for constructive interference `Delta x = m lambda`
`(4)/(3)sqrt(x^(2) + d^(2)) - sqrt(x^(2) + d^(2)) = m lambda`
`(1)/(3)sqrt(x^(2) + d^(2)) = m lambda`
`x^(2) = 9 m^(2) lambda^(2) - d^(2)`
On comparing with `x^(2) = p^(2)m^(2) lambda^(2) - d^(2)`
we get, `p = 3`