Question

An interference is observed due to two coherent sources $S_1$ placed at origin and $S_2$ placed at $(0,3\lambda ,0)$. 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 $(\text{excluding}x=0\text{and}x=\infty )$ where maximum intensity is observed.

• A. $x=2.50\lambda$
• B. $x=1.25\lambda$
• C. $x=4\lambda$
• D. $x=6\lambda$

Answer 1

Answer: (B), (C)

$x=4\lambda$

At $x=0,$ path difference $\Delta x=3\lambda$. Hence, third order maxima will be obtained.

$Atx=\infty$, path difference $\Delta x=0$, Hence, zero order maxima will be obtained.

In between $x=0\text{and}x=\infty$, first and second order maxima will be obtained.


Let the first order maxima is obtained at $Q$ and the second order maxima is obtained at $P$.

Condition for First order maxima.
$\Delta x=S_{2}Q-S_{1}Q=\lambda$

$\Rightarrow \sqrt{x_1^2+9{\lambda }^2}-x_1=\lambda$

$\Rightarrow \sqrt{x_1^2+9{\lambda }^2}=x_1+\lambda$

Squaring on both sides, we get
$x_1^2+9{\lambda }^2=x_1^2+{\lambda }^2+2x_1\lambda$

$\Rightarrow \lambda (x_1-4\lambda )=0$

$\therefore x_1=4\lambda$

Condition for second order maxima,

$\Delta x=S_{2}P-S_{1}P=2\lambda$

$\Rightarrow \sqrt{x_2^2+9{\lambda }^2}-x_2=2\lambda$

$\Rightarrow \sqrt{x_2^2+9{\lambda }^2}=x_2+2\lambda$

Squaring on both sides, we get
$x_2^2+9{\lambda }^2=x_2^2+4{\lambda }^2+4x_2\lambda$

$\therefore x_2=\frac{5}{4}\lambda =1.25\lambda$

Hence, option $\left(B\right)$ and $\left(C\right)$ are correct.