A plane electromagnetic wave in vacuum (no charges

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electromagnetic waves and their propagation · Electromagnetic waves and their propagation

Electricity, Magnetism and Electromagnetism, Ch Plane electromagnetic waves: polarisation, energy flux and radiation pressure

Problem

A plane electromagnetic wave in vacuum (no charges, no currents) travels along +y+y with E=E0cos⁡(ky−ωt)z^\mathbf{E} = E_0\cos(ky - \omega t)\hat{\mathbf{z}} and E0=60 V m−1E_0 = 60\ \mathrm{V\,m^{-1}}. Recall the energy densities 12ε0E2\tfrac12\varepsilon_0E^2 (Chapter 3) and B2/2μ0B^2/2\mu_0 (Chapter 12), and that the time average of cos⁡2\cos^2 over a period is 12\tfrac12. Take ε0=8.854×10−12 F m−1\varepsilon_0 = 8.854\times10^{-12}\ \mathrm{F\,m^{-1}} and μ0=4π×10−7 T m A−1\mu_0 = 4\pi\times10^{-7}\ \mathrm{T\,m\,A^{-1}}. ((a)) Compute the wave speed 1/μ0ε01/\sqrt{\mu_0\varepsilon_0} and hence the magnetic field amplitude B0B_0, and state the direction of B\mathbf{B}. ((b)) Find the time-averaged electric energy density ⟨uE⟩\langle u_E\rangle and the time-averaged magnetic energy density ⟨uB⟩\langle u_B\rangle, and comment on their sizes. ((c)) Find the total time-averaged energy density ⟨u⟩\langle u\rangle and, from it, the intensity I=⟨u⟩cI = \langle u\rangle c.

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