Question

The separation between two coherent point sources $S_1$ and $S_2$ vibrating in phase creating light of wavelength $\lambda$ is $2\lambda$. The smallest distance from $S_1$ on a line passing through S. and perpendicular to line $S_1{S}_2$,where a minimum of intensity occurs is $\frac{n\lambda }{12}$ where is an integer .Find the value of n then

• A. 45
• B. 66
• C. 77
• D. 87

Answer 1

The correct option is A 45

The setup in lines as follow :-

Given d=2\lambda.

Let first interference minima forms at $A$.

As path $AS,=x$ and $A{S}_2={\sqrt{x}}^2+d^2$

Path different at A will be :-

$△x={\sqrt{x}}^2+d^2-x$.

$△x={\sqrt{4x}}^2+x^2-x$

A minimum would be formed at $A$ if.

$△x=(2x-1)\frac{\lambda }{2}$ [condition for destruction interference]

$\Rightarrow {\sqrt{4\lambda }}^2+x^2-x=(2x-1)\frac{\lambda }{2}$

For first minima, $n=1$

$7{\sqrt{4\lambda }}^2{x}^2=\frac{\lambda }{2}+x$

square both sides, $\Rightarrow 4{\lambda }^2+x^2=\frac{{\lambda }^2}{4}+x^{\lambda }+x^2$

$\Rightarrow x=4\lambda -\frac{\lambda }{4}=\frac{15\lambda }{4}=\frac{45\lambda }{12}$

$\therefore n=45$