Introduction to Electrodynamics · Problem 9.35

Potentials and Fields — David J. Griffiths

Physics · spherical-wave

Problem

Suppose E(r,θ,ϕ,t)=Asin⁡θr[cos⁡(kr−ωt)−(1/kr)sin⁡(kr−ωt)]ϕ^,with ωk=c.\mathbf{E}(r,\theta,\phi,t) = A\frac{\sin\theta}{r}\left[\cos(kr - \omega t) - (1/kr)\sin(kr - \omega t)\right]\hat{\boldsymbol{\phi}}, \quad \text{with } \frac{\omega}{k} = c. (This is, incidentally, the simplest possible spherical wave. For notational convenience, let (kr−ωt)≡u(kr - \omega t) \equiv u in your calculations.) (a) Show that E\mathbf{E} obeys all four of Maxwell's equations, in vacuum, and find the associated magnetic field. (b) Calculate the Poynting vector. Average S\mathbf{S} over a full cycle to get the intensity vector I\mathbf{I}. (Does it point in the expected direction? Does it fall off like r−2r^{-2}, as it should?) (c) Integrate I⋅da\mathbf{I} \cdot d\mathbf{a} over a spherical surface to determine the total power radiated. [Answer: 4πA2/3μ0c4\pi A^2/3\mu_0 c]

Browse the problem library