Question

Two coherent sources $S_1$ and $S_2$ which emit sound of wavelength $\lambda$ in phase are separated by $2\lambda$. A circular wire of large radius is placed in such a way that $S_1{S}_2$ lies in its plane and the middle point of $S_1{S}_2$ is at the centre of the wire as shown below. Find the angular position $\theta$ (in first quadrant) on the wire for which destructive interference takes place.
[Assume the radius of ring $R>>2\lambda$]

• A. ${\cos }^{-1}\left(\frac{1}{4}\right)$
• B. ${\cos }^{-1}\left(\frac{3}{4}\right)$
• C. ${\cos }^{-1}\left(\frac{9}{4}\right)$
• D. ${\cos }^{-1}\left(\frac{11}{4}\right)$

Answer 1

Answer: (A), (B)

${\cos }^{-1}\left(\frac{3}{4}\right)$

For destructive interference
$\Delta x=(2n+1)\frac{\lambda }{2}$
From the given figure,


$\Rightarrow S_{1}P-S_{2}P=(2n+1)\frac{\lambda }{2}\left(i\right)$
In $\Delta S_{1}M{S}_2$
$\cos \alpha =\frac{S_{1}M}{S_1{S}_2}$
$\Rightarrow S_{1}M=S_1{S}_2\cos \alpha =2\lambda \cos \alpha$
Substitute in equation $\left(i\right)$
$S_{1}P-S_{2}P=S_{1}M=(2n+1)\frac{\lambda }{2}$
$\Rightarrow 2\lambda \cos \alpha =(2n+1)\frac{\lambda }{2}$
$\Rightarrow \cos \alpha =\frac{2n+1}{4}$
So, $\cos \alpha =\frac{1}{4},\frac{3}{4}$
[since $\cos \alpha \le 1$]
or $\alpha ={\cos }^{-1}\left(\frac{1}{4}\right),{\cos }^{-1}\left(\frac{3}{4}\right)$
Since ring has a large radius (i.e $R>>2\lambda$)
so $\theta =\alpha ={\cos }^{-1}\left(\frac{1}{4}\right),{\cos }^{-1}\left(\frac{3}{4}\right)$
Thus, option (a) and (b) are correct answers.