Question

The separation between two coherent point sources is $3\lambda$. On a line perpendicular to $S_1{S}_2$ and passing through $S_2$. Find the smallest distance where minimum of intensity occurs:

• A. $\frac{11\lambda }{20}$
• B. $\frac{\lambda }{2}$
• C. $\frac{9\lambda }{20}$
• D. $\frac{35\lambda }{4}$

Answer 1

The correct option is A $\frac{11\lambda }{20}$

From the above,

$d=3\lambda$

Condition for minima,

$\Delta x=(n+1)\frac{1}{\lambda }$

For $n=3$

$\Delta x=\frac{5\lambda }{2}$. . . . . . . .(1)

Path difference, $S_{1}P-S_{2}P=\frac{5}{2}\lambda$

$\frac{5}{2}\lambda =\sqrt{(d{)}^2+y^2}-y$

$\frac{5}{2}\lambda =\sqrt{(3\lambda {)}^2+y^2}-y$

$\sqrt{(3\lambda {)}^2+y^2}=\frac{5}{2}\lambda +y$

$(3\lambda {)}^2+y^2=(\frac{5}{2}\lambda +y{)}^2$

$y^2+9{\lambda }^2=\frac{25}{4}{\lambda }^2+y^2+5\lambda y$

$9{\lambda }^2-\frac{25}{4}{\lambda }^2=5\lambda y$

$5y=\frac{36\lambda -25\lambda }{4}$

$5\lambda =\frac{11}{4}\lambda$

$y=\frac{11}{20}\lambda$

The correct option is A.