Question

Ratio of intensities in consecutive maxima in a diffraction pattern due to a single slit is

• A. $1:2:3$
• B. $1:4:9$
• C. $1:\frac{2}{{\pi }^2}:\frac{3}{{\pi }^2}$
• D. $1:\frac{4}{9{\pi }^2}:\frac{4}{25{\pi }^2}$

Answer 1

The correct option is D $1:\frac{4}{9{\pi }^2}:\frac{4}{25{\pi }^2}$

The intensity is given by:

$I=I_0\left(\frac{sin\varphi /2}{\varphi /2}\right)$

For nth secondary maxima, $dsin\theta =\left(\frac{2n+1}{2}\right)\lambda$

$\therefore \frac{\varphi }{2}=\frac{\pi }{2}\left[dsin\theta \right]=\left(\frac{2n+1}{2}\right)\pi$

So, $I=I_0{\left(\frac{sin\left(\frac{2n+1}{2}\right)\pi }{\left(\frac{2n+1}{2}\right)\pi }\right)}^2=\frac{I_0}{{\left[\left(\frac{2n+1}{2}\right)\pi \right]}^2}$

So, $I_0:I_1:I_2=I_0:\frac{4}{9{\pi }^2}{I}_0:\frac{4}{25{\pi }^2}{I}_0$

So, ratio is $1:\frac{4}{9{\pi }^2}:\frac{4}{25{\pi }^2}$