Answer: 9

Solution

In young’s double slit experiment, intensity at a point is given by

$I = I_0\cos^2\frac{\phi}{2}$ …(i)

where, $\phi$ = phase difference,

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),

$\frac{I_1}{I_2} = \frac{\cos^2\left(\frac{\phi_1}{2}\right)}{\cos^2\left(\frac{\phi_2}{2}\right)} = \frac{\cos^2(2\pi)}{\cos^2\left(\frac{\pi}{3}\right)}$

$\Rightarrow \frac{K}{I_2} = \frac{1}{4/3} = \frac{4}{3} \Rightarrow I_2 = \frac{3K}{4} = \frac{9K}{12}$

$\therefore n = 9$