Question

Consider the optical system shown in the figure that follows. The point source of light $S$ is having wavelength equal to $\lambda$. The light is reaching screen only after reflection. For point $P$ to be $2^{nd}$ maxima, the value of $\lambda$ would be $(D>>d$ and $d>>\lambda )$ :

• A. $\frac{12d^2}{D}$
• B. $\frac{6d^2}{D}$
• C. $\frac{3d^2}{D}$
• D. $\frac{24d^2}{D}$

Answer 1

The correct option is A $\frac{12d^2}{D}$

Two rays from light source S get reflected from the two mirrors at M and N respectively and then interfere at point P.

Length of path $SMP=\sqrt{d^2+(\frac{D}{2}{)}^2}+\frac{D}{2}$

Length of path $SNP=\sqrt{(3d{)}^2+(\frac{D}{2}{)}^2}+\sqrt{(\frac{D}{2}{)}^2+(4d{)}^2}$

For second maxima to be obtained at P,the path difference must be $2\lambda$

i-e $SNP-SMP=$ $[\sqrt{(3d{)}^2+(\frac{D}{2}{)}^2}+\sqrt{(\frac{D}{2}{)}^2+(4d{)}^2}]-$$[\sqrt{d^2+(\frac{D}{2}{)}^2}+\frac{D}{2}]=2\lambda$

Given d<<D

using Bionomial Expansion, $\sqrt{1+x}=1+\frac{x}{2}forx<<1$

$\lambda =\frac{12d^2}{D}$