Chapter 24, Problem 31P

To determine

The angle at which the third order be observed.

Answer to Problem 31P

  $53.3°$

Explanation of Solution

Given info:

Wavelength of the light, $\lambda =589nm=589\times {10}^{-9}m$

Angle, $\theta =15.5°$

First order, $m=1$

Let the distance between the slits be $d$ .

Formula used:

  $\sin \theta =\frac{m\lambda }{d}$

Where,

  $m$ is an integer.

  $\lambda$ is wavelength.

  $d$ is distance between slits.

Calculation:

The path difference is an integer multiple of the wavelength in constructive interference.

Rearrange the given formula: $\sin \theta =\frac{m\lambda }{d}$ . So, $d\sin \theta =m\lambda$

First calculate the slit separation from by using $d\sin \theta =m\lambda$

Substitute the given values:

  $d\sin 15.5°=\left(1\right)\left(589\times {10}^{-9}m\right)$

This gives, $d=2.20\times {10}^{-6}m$

Now calculate the angle for the third order for by using $\theta ={\sin }^{-1}\frac{3\lambda }{d}$

Substitute the given values:

  $\theta ={\sin }^{-1}\frac{3\left(589\times {10}^{-9}m\right)}{\left(2.20\times {10}^{-6}m\right)}=53.3°$

Conclusion:

The required angle is $53.3°$ .