Chapter 24, Problem 21Q

To determine

To Explain: The orders that will be overlap in the observed spectrum. Whether the answer will depend on the slit spacing.

Explanation of Solution

Introduction: The pattern resulting from superimposing of two waves is known as interference. Interference can be destructive or constructive.

Destructive interference occurs when the crest of a wave overlaps the trough of another wave. The path difference is an integer multiple of the wavelength in constructive interference. So, $d\sin \theta =m\lambda$

Where,

  $m$ is an integer

  $\lambda$ is wavelength

  $d$ is a distance

  $\theta$ is the angle relative to original direction of light,

According to this, for red colour & violet colours from different orders to overlap. So, angles in the equation will be $d\sin \theta =m\lambda$

Mathematically, $\frac{\left(m+1\right)\left(400nm\right)}{d}<\frac{m\left(700nm\right)}{d}$

Therefore, $m^{th}$ order $700nm$ line overlaps ${\left(m+1\right)}^{th}$ order $400nm$ , experiential spectra will always overlap at this as well as higher orders.

So, the answer will not depend on $d$ and slit spacing will also cancel out.

Conclusion:  Therefore, $m^{th}$ order $700nm$ line overlaps ${\left(m+1\right)}^{th}$ order $400nm$ , experiential spectra will always overlap at this as well as higher orders.