ElectromagneticWavesElectromagneticMomentumRadiationPressureAndPhotonMomentumExercises 2026-09-19 19:41:04 2026-09-19 19:41:04 Topic adding in sections a bit at a time since it initially failed to save 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 % this is the default PhysicsLibrary preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics). % If you use \includegraphics{figure} with no file extension, PhysicsLibrary % will prefer figure.png before falling back to figure.eps. \usepackage{graphicx} \DeclareGraphicsExtensions{.png,.jpg,.jpeg,.pdf,.eps} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \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}. \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}