Question

Two coherent sources $S_1$ and $S_2$ separated by distance $2\lambda$ emit light of wavelength $\lambda$ in phase as shown in the figure. A circular wire of radius $100\lambda$ is placed in such a way that $S_1{S}_2$ lies in its plane and the mid-point of $S_1{S}_2$ is at the centre of wire.

(i) Find the angular positions $\theta$ on the wire for which constructive interference takes place. Hence or otherwise find the total number of maxima.

(ii) Find the angular positions $\theta$ (in the first quadrant) on the wire for which intensity reduces to half of its maximum value.

Answer 1

$\frac{I_{max}}{2}=I=I_0+I_0+2\sqrt{I_0}\sqrt{I_0}cos\theta$

$\frac{4I_0}{2}=2I_0+2I_{0}cos\theta$
get $cos\varphi =0$

$\varphi =\frac{\pi }{2}\Rightarrow \Delta =\left(\frac{\lambda }{2\pi }\right)\left(\frac{\pi }{2}\right)=\frac{\lambda }{4}$
to get path difference $\frac{\lambda }{4}$ we have $dcos\theta =\frac{\lambda }{4}\Rightarrow cos\theta =\frac{1}{8}$

$\Rightarrow 32cos\theta =4$