Question

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

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

Answer 1

The correct option is A $\frac{d^2}{D}$


Path difference at distance $y$ from point $O$ is given as,
$\Delta x=\frac{yd}{D}$
But $y=\frac{d}{2}$
$\therefore \Delta x=\frac{d^2}{2D}$

"The wavelength will be missing" means that a dark band will form corresponding to each "missing wavelengths".

So, for missing wavelengths, we can write,
$\Delta x=\frac{{\lambda }_1}{2},\frac{3{\lambda }_2}{2},\frac{5{\lambda }_3}{2}...$ and so on

$\Rightarrow {\lambda }_1=2\Delta x=\frac{d^2}{D}$

$\Rightarrow {\lambda }_2=\frac{2\Delta x}{3}=\frac{d^2}{3D}$

$\Rightarrow {\lambda }_3=\frac{2\Delta x}{5}=\frac{d^2}{5D}$

And, so on

So, the missing wavelengths are,

$\frac{d^2}{D},\frac{d^2}{3D},\frac{d^2}{5D}........$and so on

Hence, $\left(A\right)$ is the correct answer.

Why this question? Tip : Wavelength in interference pattern is missing when path difference between interfering light is, $\Delta x=\left(\frac{2n-1}{2}\right)\lambda$, where, $n=1,2,\mathrm{3......}$