Introduction to Electrodynamics · Problem 11.26

Electrodynamics and Relativity — David J. Griffiths

Physics · electric-dipole-radiation

Problem

An ideal electric dipole is situated at the origin; its dipole moment points in the

\hat{\mathbf{z}}

direction, and is quadratic in time:

\mathbf{p}(t) = \frac{1}{2}\ddot{p}_0 t^2\,\hat{\mathbf{z}}, \quad (-\infty < t < \infty)

where

\ddot{p}_0

is a constant. (a) Use the method of Section 11.1.2 to determine the (exact) electric and magnetic fields, for all

r > 0

(there's also a delta-function term at the origin, but we're not concerned with that).

\left[Partial\ Answer: V = \frac{\mu_0 \ddot{p}_0}{8\pi}\cos\theta[(ct/r)^2 - 1],\ \mathbf{A} = \frac{\mu_0 \ddot{p}}{4\pi c}[(ct/r) - 1]\hat{\mathbf{z}}.\right]

(b) Calculate the power,

P(r,t)

, passing through a sphere of radius

r

\left[Answer: \frac{\ddot{p}_0^2}{12\pi\epsilon_0 r^3}t[t^2 + (r/c)^2].\right]