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 $nI_{0}$ , then what is the value of $\left[n\right]$ ? Here $\left[.\right]$ is the greatest integer function.

Answer 1

$\Delta x=\frac{\lambda }{8}$
$\phi \, $ between 1 and 3 $=\frac{2 \pi \, }{\lambda }\times \frac{\lambda }{4}=\frac{\pi }{2}$
$\phi$ between 2 and 4 $=\frac{\pi }{2}$
Intensity because of 1 and 3, 2 and 4
$=I_{1}+I_{2}+2\sqrt{I_{1} I_{2}} \, cos\phi$ = $I_{0}+I_{0}+2\sqrt{I_{0} I_{0}}cos\left(\right.\left(\pi \right)/2\left.\right)$
$=2I_{0}$
Now phase difference between resultant of 1 and 3, 2 and 4 is $\frac{\pi }{4}$
$\therefore \, \, I=I_{1}+ \, \, I_{2}+2\sqrt{I_{1}}, \, \sqrt{I_{2}}cos d$
$=2I_{0}+2I_{0}+2\sqrt{2}\times \sqrt{2}I_{0}\times \frac{1}{\sqrt{2}}$
$4I_{0}+2\sqrt{2}I_{0}$
$\left(4 I_{0} + 2 \sqrt{2}\right)I_{0}$
$=6.8 \, I_{0}$
$\therefore \, \, \left[n\right]=6$