A light source emits a beam of photons, each of which has a momentum of

$2.3 \times 10 ^ { - 29 } \mathrm { kg } \cdot \mathrm { m } / \mathrm { s }$

(a) What is the frequency of the photons? (b) To what region of the electromagnetic spectrum do the photons belong?

Solution

VerifiedThe momentum of a photon is given by:

$\begin{align} p = \frac{h}{\lambda} \end{align}$

where $h$ is Plank's constant and $\lambda$ the wavelength of the photon.

Frequency and wavelength are connected via the following equation:

$\begin{align} c = \lambda\nu \end{align}$

where $c$ corresponds to the speed of light in vacuum. Use equation (2) to express $\lambda$ in terms of the speed of light and the frequency:

$\begin{align*} \lambda = \frac{c}{\nu} \end{align*}$

Now, put this into (1):

$\begin{align*} p = \frac{\nu h}{c}\hspace{0.2cm}\Rightarrow\hspace{0.2cm}\nu & = \frac{pc}{h}\\&= \frac{3\cdot10^8\mathrm{m}/\mathrm{s}\cdot2.3\cdot10^{-29}\mathrm{kgm/s}}{6.63\cdot10^{-34}\mathrm{Js}}\\ & = \boxed{1.04\cdot10^{13}\mathrm{Hz}} \end{align*}$

This photon, according to Figure 24.9 in the textbook, lies in the $\textit{infrared}$ region of the electromagnetic spectrum