Question

Consider four monochromatic and coherent sources of light-emitting waves in phase when placed on $y$ axis at $y=0,a,2a$ and $3a.$ The intensity of wave reaching point $P$ far away on $y$ axis from each of the sources is almost the same and equal to $I_0$ . If it is found that the resultant intensity at $P$ for $a=\frac{\lambda }{8}$ is $n{I}_0$ , then what is the value of $\left[n\right]$ ? Here $\left[.\right]$ is the greatest integer function.

Answer 1

Answer: 6

$\Delta x=\frac{\lambda }{8}$
$\varphi$ between 1 and 3 $=\frac{2\pi }{\lambda }\times \frac{\lambda }{4}=\frac{\pi }{2}$
$\varphi$ between 2 and 4 $=\frac{\pi }{2}$
Intensity because of 1 and 3, 2 and 4
$=I_1+I_2+2\sqrt{I_1{I}_2}cos\varphi$ = $I_0+I_0+2\sqrt{I_0{I}_0}cos(\left(\pi \right)/2)$
$=2I_0$
Now phase difference between resultant of 1 and 3, 2 and 4 is $\frac{\pi }{4}$
$\therefore I=I_1+I_2+2\sqrt{I_1},\sqrt{I_2}cosd$
$=2I_0+2I_0+2\sqrt{2}\times \sqrt{2}{I}_0\times \frac{1}{\sqrt{2}}$
$4I_0+2\sqrt{2}{I}_0$
$\left(4I_0+2\sqrt{2}\right)I_0$
$=6.8I_0$
$\therefore \left[n\right]=6$