Question

In the given figure $A$ and $B$ are identical radiators of wave that are in phase and are of same wavelength $\lambda$. The radiators are separated by a distance $d=3.00\lambda$, the greatest distance from $A$, along the $x-axis$ for which fully destructive interference occurs. The distance in terms of wavelength is

• A. $8.75\lambda$
• B. $2.5\lambda$
• C. $4.5\lambda$
• D. $11.25\lambda$

Answer 1

The correct option is A $8.75\lambda$

Let the point along x axis be P.

The path lengths will be PA and PB.

Path difference $\Delta =PB-PA=\sqrt{x^2+d^2}-x$

For destructive interference, $\Delta =(n+\frac{1}{2})\lambda$

$\therefore \sqrt{x^2+d^2}-x=(n+\frac{1}{2})\lambda$

$\therefore x^2+d^2=(x+(n+\frac{1}{2})\lambda {)}^2$

For farthest minima, $n=0$

$\Rightarrow d^2=x\lambda +\frac{{\lambda }^2}{4}$

$\Rightarrow x=\frac{d^2}{\lambda }-\frac{\lambda }{4}=\frac{9{\lambda }^2}{\lambda }-\frac{\lambda }{4}=\frac{35\lambda }{4}=8.75\lambda$