Question

Monochromatic light of wavelength '$\lambda$' passes through a pair of slits $S_1$ and $S_2$ separated by a distance equal to fringe width $\beta$. There is a screen placed at a distance $D$ in which two parallel slits $S_1$ and $S_2$ are cut at the positions of 3rd and 4th maxima. Again there is a screen at a distance $D$.

P is a point at a distance of $\frac{\beta }{2}$ from the central line of the first pair of slits $S_1$ and $S_2$ , while Q is on the central line of the second pair of slits $S_1$ and $S_2$ All slits are of same widths.

If the intensity of light at the point P is I and the intensity at $Q$ is $\frac{xI}{4}$ then find the value of '$x$'.

Answer 1

For slit $S_4$, path difference is $4\lambda ⟹\delta =\frac{2\pi }{\lambda }\times 4\lambda =8\pi$

Intensity of slit $I_4=4I_{o}co{s}^2\frac{8\pi }{2}$, where $I_o$ is the intensity of the slits $S_1$ and $S_2$

$I_4=4I_o$

For slit $S_3$, path difference is $3\lambda ⟹\delta =\frac{2\pi }{\lambda }\times 3\lambda =6\pi$

Intensity of slit $I_3=4I_{o}co{s}^2\frac{6\pi }{2}$

$I_3=4I_o$

Now, as $QP=3\beta$ i-e, P is situated at $3rd$ maxima for slits $S_3$ and $S_4$.

Thus intensity at P, $I=4\left(4I_o\right)co{s}^2\frac{6\pi }{2}=16I_o$

$I_Q=4\left(4I_o\right)co{s}^2{0}^o$ (as path difference is 0 $⟹\delta =0$ )

Hence, $I_Q=16I_o=I$

According to question, $I=\frac{xI}{4}⟹x=4$