Classical Electrodynamics · Problem 9.6

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · electric-dipole-arbitrary-time-dependence

Problem

(a) Starting from the general expression (9.2) for A\mathbf{A} and the corresponding expression for Φ\Phi, expand both R=∣x−x′∣R = |\mathbf{x} - \mathbf{x}'| and t′=t−R/ct' = t - R/c to first order in ∣x′∣/r|\mathbf{x}'|/r to obtain the electric dipole potentials for arbitrary time variation Φ(x, t)=14πϵ0[1r2 n⋅pret+1cr n⋅∂pret∂t]A(x, t)=μ04πr ∂pret∂t\begin{aligned} \Phi(\mathbf{x},\, t) &= \frac{1}{4\pi\epsilon_0}\left[\frac{1}{r^2}\, \mathbf{n} \cdot \mathbf{p}_{\text{ret}} + \frac{1}{cr}\, \mathbf{n} \cdot \frac{\partial \mathbf{p}_{\text{ret}}}{\partial t}\right] \\ \mathbf{A}(\mathbf{x},\, t) &= \frac{\mu_0}{4\pi r}\, \frac{\partial \mathbf{p}_{\text{ret}}}{\partial t} \end{aligned} where pret=p(t′=t−r/c)\mathbf{p}_{\text{ret}} = \mathbf{p}(t' = t - r/c) is the dipole moment evaluated at the retarded time measured from the origin. (b) Calculate the dipole electric and magnetic fields directly from these potentials and show that B(x, t)=μ04π[−1cr2 n×∂pret∂t−1c2r n×∂2pret∂t2]E(x, t)=14πϵ0{(1+rc∂∂t)[3n(n⋅pret)−pretr3]+1c2r n×(n×∂2pret∂t2)}\begin{aligned} \mathbf{B}(\mathbf{x},\, t) &= \frac{\mu_0}{4\pi}\left[-\frac{1}{cr^2}\, \mathbf{n} \times \frac{\partial \mathbf{p}_{\text{ret}}}{\partial t} - \frac{1}{c^2 r}\, \mathbf{n} \times \frac{\partial^2 \mathbf{p}_{\text{ret}}}{\partial t^2}\right] \\ \mathbf{E}(\mathbf{x},\, t) &= \frac{1}{4\pi\epsilon_0}\left\{\left(1 + \frac{r}{c}\frac{\partial}{\partial t}\right)\left[\frac{3\mathbf{n}(\mathbf{n} \cdot \mathbf{p}_{\text{ret}}) - \mathbf{p}_{\text{ret}}}{r^3}\right] + \frac{1}{c^2 r}\, \mathbf{n} \times \left(\mathbf{n} \times \frac{\partial^2 \mathbf{p}_{\text{ret}}}{\partial t^2}\right)\right\} \end{aligned} (c) Show explicitly how you can go back and forth between these results and the harmonic fields of (9.18) by the substitutions −iω↔∂/∂t-i\omega \leftrightarrow \partial/\partial t and peikr−iωt↔pret(t′)\mathbf{p}e^{ikr-i\omega t} \leftrightarrow \mathbf{p}_{\text{ret}}(t').

Related theory

Browse the problem library