Question

White light is used to illuminate two slits in Young's double slit experiment. The separation between the slits is b and the screen is at a distance $d(>>b)$ from the slits. At a point on the screen directly in front of one of the slits, which wavelengths are missing?

• A. $\frac{b}{d},\frac{b}{3d},\frac{b}{5d}$
• B. $\frac{b^2}{2d},\frac{b^2}{4d},\frac{b^2}{6d}$
• C. $\frac{b^2}{d},\frac{b^2}{3d},\frac{b^2}{5d}$
• D. $\frac{b}{2d},\frac{b}{4d},\frac{b}{6d}$

Answer 1

Answer: (C)

$\frac{b^2}{d},\frac{b^2}{3d},\frac{b^2}{5d}$

The situation is shown in the figure.
The path difference between the waves reaching the point P is
$\Delta p=BP-AP$
$=(d^2-b^2{)}^{1/2}-d=d{(1+\frac{b^2}{d^2})}^{1/2}-d$
$=d\left[1+\frac{1}{2}\frac{b^2}{d^2}\right]-d=\frac{b^2}{2d}$
For a dark band
$\Delta p=\frac{b^2}{2d}=(2n-1)\frac{\lambda }{2}$
where, $n=1,2,3$ or $\lambda =\frac{1}{(2n-1)}\cdot \frac{b^2}{d}$
Hence, the missing wavelengths are $\frac{b^2}{d},\frac{b^2}{3d},\frac{b^2}{5d}$.