Electromagnetic Waves: Electromagnetic Momentum, Radiation Pressure, and Photon Momentum - Exercises
ElectromagneticWavesElectromagneticMomentumRadiationPressureAndPhotonMomentumExercises
2026-09-19 19:41:04
2026-09-19 19:42:20
Topic
more sections, maybe it was the long title
electromagnetic momentum
momentum density
Poynting vector
Maxwell stress tensor
radiation pressure
absorption
reflection
transmission
oblique incidence
solar sail
photon momentum
photon flux
optical force
RF momentum transfer
exercises
worked solutions
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\section*{Electromagnetic Waves, Antennas, and RF: Electromagnetic Momentum, Radiation Pressure, and Photon Momentum - Exercises and Complete Worked Solutions}
EM18 established that electromagnetic fields carry momentum as well as energy. This companion article develops that result through worked problems ranging from local field momentum density to radiation pressure, the Maxwell stress tensor, solar-sail acceleration, photon momentum, and the agreement between classical and quantum momentum accounting.
The central vacuum relations are
\begin{equation}
\boxed{\mathbf g=\epsilon_0\mathbf E\times\mathbf B=\frac{\mathbf S}{c^2},}
\end{equation}
\begin{equation}
\boxed{\mathbf P_{\text{EM}}=\int_V\mathbf g\,dV,}
\end{equation}
and, for a plane wave,
\begin{equation}
\boxed{g=\frac{u}{c},\qquad P_{\text{EM}}=\frac{U}{c}.}
\end{equation}
At normal incidence, the radiation-pressure limits are
\begin{equation}
\boxed{p_{\text{abs}}=\frac{I}{c},\qquad p_{\text{refl}}=\frac{2I}{c}.}
\end{equation}
The photon description uses
\begin{equation}
\boxed{E_\gamma=h\nu=\frac{hc}{\lambda},\qquad p_\gamma=\frac{E_\gamma}{c}=\frac{h}{\lambda}.}
\end{equation}
The problems below are intended to make these formulas consequences of momentum conservation rather than isolated facts \cite{Griffiths2017,Jackson1999,OpenStaxV2,FeynmanV1,FeynmanV2}.
\section*{How to use this problem set}
Attempt all exercises in Part I before consulting Part II. In every radiation-pressure problem, identify three things before calculating: the direction of the incident momentum, what fraction of that momentum leaves after interaction, and whether the requested force is along the beam direction or normal to a material surface. This prevents most sign and factor-of-two errors.
\begin{center}
\includegraphics{EM18E1_fig01_momentum_chain.png}
\vspace{0.45em}
\textbf{Figure.} The electromagnetic momentum chain. The fields determine the Poynting vector, the Poynting vector determines momentum density, and momentum flux produces force and pressure when the radiation interacts with matter.
\end{center}
\section*{Part I: Exercises}
\section*{Exercise 1: momentum density of a vacuum plane wave}
At a particular point and instant, a vacuum plane wave has
\begin{equation}
\mathbf E=(300\,\text{V/m})\hat{\mathbf x}
\end{equation}
and propagates in the $+z$ direction.
Find:
\begin{enumerate}
\item[(a)] the corresponding magnetic-field vector $\mathbf B$;
\item[(b)] the instantaneous Poynting vector $\mathbf S$;
\item[(c)] the electromagnetic momentum density $\mathbf g$ using $\mathbf g=\mathbf S/c^2$;
\item[(d)] the total instantaneous energy density $u$;
\item[(e)] verify numerically that $g=u/c$.
\end{enumerate}
\section*{Exercise 2: total momentum in a finite electromagnetic pulse}
A short vacuum pulse carries total electromagnetic energy
\begin{equation}
U=2.4\,\text{mJ}
\end{equation}
through a beam of cross-sectional area
\begin{equation}
A=3.0\,\text{cm}^2.
\end{equation}
Its duration is
\begin{equation}
\tau=8.0\,\text{ns}.
\end{equation}
Assume a uniform rectangular pulse profile.
Find:
\begin{enumerate}
\item[(a)] the total electromagnetic momentum of the pulse;
\item[(b)] the pulse length $L=c\tau$;
\item[(c)] the pulse volume $V=AL$;
\item[(d)] the average energy density $u=U/V$ inside the pulse;
\item[(e)] the average momentum density $g=u/c$;
\item[(f)] verify that $gV=U/c$.
\end{enumerate}
\section*{Exercise 3: absorbing surface under a known intensity}
A normally incident electromagnetic beam has intensity
\begin{equation}
I=1200\,\text{W/m}^2
\end{equation}
and completely illuminates an absorbing plate of area
\begin{equation}
A=0.35\,\text{m}^2.
\end{equation}
Find:
\begin{enumerate}
\item[(a)] the radiation pressure;
\item[(b)] the force on the plate;
\item[(c)] the momentum transferred to the plate during $10\,\text{s}$.
\end{enumerate}
\section*{Exercise 4: perfect mirror and the factor of two}
A $8.0\,\text{W}$ laser beam is completely intercepted by a perfect mirror at normal incidence.
Find:
\begin{enumerate}
\item[(a)] the incident electromagnetic momentum arriving per second;
\item[(b)] the reflected electromagnetic momentum leaving per second, including its direction;
\item[(c)] the force on the mirror;
\item[(d)] explain from momentum conservation why the result is twice the absorbing-surface force for the same beam power.
\end{enumerate}
\section*{Exercise 5: partial absorption, reflection, and transmission}
A plane wave of intensity
\begin{equation}
I=2500\,\text{W/m}^2
\end{equation}
strikes a planar optical element at normal incidence. The power fractions are
\begin{equation}
A=0.25,
\qquad
R=0.60,
\qquad
T=0.15.
\end{equation}
Assume the incident, reflected, and transmitted beams are all in vacuum, with transmission continuing in the original direction.
Find:
\begin{enumerate}
\item[(a)] the incoming momentum flux;
\item[(b)] the outgoing reflected and transmitted momentum fluxes, with signs;
\item[(c)] the radiation pressure on the element;
\item[(d)] verify the equivalent formula
\begin{equation}
p_{\text{rad}}=\frac{(A+2R)I}{c}.
\end{equation}
\end{enumerate}
\begin{center}
\includegraphics{EM18E1_fig02_partial_surface_momentum.png}
\vspace{0.45em}
\textbf{Figure.} Momentum bookkeeping for a surface that absorbs, reflects, and transmits portions of an incident wave. The reflected momentum reverses sign, which is why reflection contributes twice its fractional power to the pressure.
\end{center}
\section*{Exercise 6: oblique reflection}
Sunlight with intensity
\begin{equation}
I=1360\,\text{W/m}^2
\end{equation}
strikes a perfectly reflecting flat sail of actual area
\begin{equation}
A=20\,\text{m}^2
\end{equation}
at an angle
\begin{equation}
\theta=35^\circ
\end{equation}
measured from the surface normal.
Find:
\begin{enumerate}
\item[(a)] the projected area seen by the beam;
\item[(b)] the incident power intercepted by the sail;
\item[(c)] the normal radiation pressure;
\item[(d)] the normal force on the sail.
\end{enumerate}
Explain physically why two factors of $\cos\theta$ appear in the normal force.
\section*{Exercise 7: Maxwell stress tensor for a plane wave}
A sinusoidal plane wave propagates in the $+z$ direction with peak electric-field amplitude
\begin{equation}
E_0=300\,\text{V/m}.
\end{equation}
At an instant when the electric field is at its positive peak, take
\begin{equation}
\mathbf E=E_0\hat{\mathbf x},
\qquad
\mathbf B=\frac{E_0}{c}\hat{\mathbf y}.
\end{equation}
Using
\begin{equation}
\sigma_{ij}
=
\epsilon_0\left(E_iE_j-\frac{1}{2}\delta_{ij}E^2\right)
+
\frac{1}{\mu_0}\left(B_iB_j-\frac{1}{2}\delta_{ij}B^2\right),
\end{equation}
find:
\begin{enumerate}
\item[(a)] the instantaneous total energy density $u$;
\item[(b)] $\sigma_{zz}$ at the field peak;
\item[(c)] the cycle-averaged magnitude $\langle|\sigma_{zz}|\rangle$;
\item[(d)] verify that $\langle|\sigma_{zz}|\rangle=I/c$.
\end{enumerate}
\section*{Exercise 8: stress tensor of a static electric field}
A uniform electrostatic field is
\begin{equation}
\mathbf E=(2.0\times10^6\,\text{V/m})\hat{\mathbf x},
\qquad
\mathbf B=\mathbf 0.
\end{equation}
Find the Maxwell stress tensor in Cartesian coordinates.
Then determine the traction vector
\begin{equation}
\mathbf t=\boldsymbol{\sigma}\cdot\hat{\mathbf n}
\end{equation}
for surfaces whose outward normals are:
\begin{enumerate}
\item[(a)] $\hat{\mathbf n}=\hat{\mathbf x}$;
\item[(b)] $\hat{\mathbf n}=\hat{\mathbf y}$.
\end{enumerate}
Interpret the signs as tension along the field direction and compression transverse to the field.
\begin{center}
\includegraphics{EM18E1_fig03_maxwell_stress_cube.png}
\vspace{0.45em}
\textbf{Figure.} A static electric field produces anisotropic electromagnetic stress: tensile along the field direction and compressive on transverse faces in the stated stress-tensor sign convention.
\end{center}
\section*{Exercise 9: impulse from reflecting a finite-energy pulse}
A light pulse of energy
\begin{equation}
U=0.75\,\text{J}
\end{equation}
reflects normally from a free mirror of mass
\begin{equation}
m=2.0\,\text{g}.
\end{equation}
Assume the mirror is initially at rest and its acquired speed is sufficiently small that the change in photon energy can be neglected to first order.
Find:
\begin{enumerate}
\item[(a)] the impulse delivered to the mirror;
\item[(b)] the mirror's change in speed.
\end{enumerate}
\section*{Exercise 10: idealized solar-sail acceleration}
At approximately $1\,\text{AU}$ from the Sun, take the solar intensity to be
\begin{equation}
I=1361\,\text{W/m}^2.
\end{equation}
An ideal perfectly reflecting sail has area
\begin{equation}
A=100\,\text{m}^2
\end{equation}
and total spacecraft mass
\begin{equation}
m=12\,\text{kg}.
\end{equation}
Assume normal incidence and neglect all other forces and the variation of solar intensity with distance.
Find:
\begin{enumerate}
\item[(a)] the radiation force;
\item[(b)] the resulting acceleration;
\item[(c)] the idealized change in speed after one day.
\end{enumerate}
\section*{Exercise 11: energy and momentum of a green photon}
A photon has wavelength
\begin{equation}
\lambda=532\,\text{nm}.
\end{equation}
Find:
\begin{enumerate}
\item[(a)] the frequency;
\item[(b)] the photon energy in joules;
\item[(c)] the photon momentum;
\item[(d)] verify numerically that $E_\gamma=p_\gamma c$.
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{Griffiths2017}
David J. Griffiths,
\emph{Introduction to Electrodynamics},
4th ed., Cambridge University Press, 2017,
sections on electromagnetic momentum and the Maxwell stress tensor.
\bibitem{Jackson1999}
John David Jackson,
\emph{Classical Electrodynamics},
3rd ed., Wiley, 1999,
sections on electromagnetic conservation laws, momentum, and stress.
\bibitem{OpenStaxV2}
Samuel J. Ling, Jeff Sanny, and William Moebs,
\emph{University Physics, Volume 2},
OpenStax, 2016,
sections on electromagnetic waves, momentum, and radiation pressure.
\bibitem{FeynmanV1}
Richard P. Feynman, Robert B. Leighton, and Matthew Sands,
\emph{The Feynman Lectures on Physics, Volume I},
Addison-Wesley, 1963,
chapters on radiation, photons, and momentum transfer.
\bibitem{FeynmanV2}
Richard P. Feynman, Robert B. Leighton, and Matthew Sands,
\emph{The Feynman Lectures on Physics, Volume II},
Addison-Wesley, 1964,
chapters on electromagnetic energy, momentum, and stress.
\bibitem{Barnett2010}
Stephen M. Barnett,
``Resolution of the Abraham--Minkowski Dilemma,''
\emph{Physical Review Letters},
Vol. 104, 070401, 2010.
\end{thebibliography}