Question

When light rays approach the slits as shown in the figure below making an angle $\theta$ with the central axis, the number of maxima lying between $O$ and the central maxima will be :

• A. $\frac{d\sin \theta }{\lambda }$
• B. $\frac{d\sin \theta }{\lambda }-1$
• C. $\frac{yD}{d}$
• D. $\frac{d\sin \theta }{2\lambda }$

Answer 1

The correct option is B $\frac{d\sin \theta }{\lambda }-1$

Path difference between the rays on the screen is given by :

$\Delta x=d\sin \theta +\frac{yd}{D}$

For maxima :
$\Delta x=n\lambda$

Where, $n=0,1,2,....$

$\Rightarrow d\sin \theta +\frac{yd}{D}=n\lambda$

Or, $y=(n\lambda -d\sin \theta )\frac{D}{d}$

Hence maxima will occur at :

$y=-D\sin \theta ,\frac{\lambda D}{d}-D\sin \theta ,....$

For maxima to occur at $O$,

$y=0\Rightarrow n\lambda =d\sin \theta$

$\Rightarrow n=\frac{d\sin \theta }{\lambda }$

Hence, number of maxima between central maxima$(y=-D\sin \theta )$ and $O$ is $=\frac{d\sin \theta }{\lambda }-1$
$$\begin{array}{c}\text{Why this question?}\\ \text{Note : When light rays approaching the slit are not parallel to central axis and}\\ \text{come from above the central axis, central maxima is found below the central axis.}\end{array}$$