A young's double-slit experiment is performed using monocromatic light of wavelength $\lambda$. The inntensity of light at a point on the screen, where the path difference is $\lambda$, is $\mathrm{K}$ units. The intensity of light at a point where the path
difference is $\frac{\lambda}{6}$ is given by $\frac{\mathrm{nK}}{12}$, where $\mathrm{n}$ is an integer. The value of $n$ is_________
(9)
In young's double slit experiment, intensity at a point is given by
$I=I_{0} \cos ^{2} \frac{\phi}{2}$ ....(1)
Using phase difference, $\phi=\frac{2 \pi}{\lambda} \times$ path difference
For path difference $\lambda$, phase difference $\phi_{1}=2 \pi$
For path difference, $\frac{\lambda}{6}$, phase difference $\phi_{2}=\frac{\pi}{3}$
Using equation (i),
$\frac{I_{1}}{I_{2}}=\frac{\cos ^{2}\left(\frac{\phi_{1}}{2}\right)}{\cos ^{2}\left(\frac{\phi_{2}}{2}\right)}=\frac{\cos ^{2}\left(\frac{2 \pi}{2}\right)}{\cos ^{2}\left(\frac{\pi}{3}\right)}$
$\Rightarrow \frac{K}{I_{2}}=\frac{1}{\frac{3}{3}}=\frac{4}{3} \Rightarrow I_{2}=\frac{3 K}{4}=\frac{9 K}{12}$
$\therefore n=9$
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