Question

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

2.3×1029kgm/s2.3 \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

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The momentum of a photon is given by:

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

where hh is Plank's constant and λ\lambda the wavelength of the photon.

Frequency and wavelength are connected via the following equation:

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

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

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

Now, put this into (1):

p=νhcν=pch=3108m/s2.31029kgm/s6.631034Js=1.041013Hz\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 infrared\textit{infrared} region of the electromagnetic spectrum

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