Question

Two microwave source $S_1$ and $S_2$ are located at a distance of $4\lambda$ from each other along $y$−axis, with $S_1$ at the origin as shown in the figure. A microwave detector $D$ moves along the $x$−axis away from $S_1$. The first maximum is obtained for $(x>0)$ equal to $\frac{n\lambda }{6}$. Find $n$ (wavelength of the microwave is $\lambda$)

Answer 1

At origin, the path difference will be maximum and it equals,
$S_1-S_2=4\lambda$

• As the wave from $S_2$ travels distance $4\lambda$ wave from $S_1$ does not travel any distance.
• As we move away from origin on $x$−axis, the path difference decreases continuously.
• The nearest distance from origin where we obtain a maxima will be corresponding to path difference $3\lambda$. (for maxima, path difference = integral multiple of $\lambda )$

Let $P$ be the point of nearest maxima
path difference at$P$

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

$\Delta x=\sqrt{x^2+16{\lambda }^2}-x$

$\sqrt{x^2+16{\lambda }^2}-x=3\lambda$

$x^2+16{\lambda }^2=x^2+9{\lambda }^2+6\lambda x$

$7{\lambda }^2=6\lambda x$

$x=\frac{7\lambda }{6}$

Comparing with $\frac{n\lambda }{6}$, we get

$n=7$

FINAL ANSWER: $7$