10.2.1 Retarded Potentials #
In the Lorenz gauge, we want to find the potentials by solving
\[\laplacian \vec A - \mu_0 \epsilon_0 \pdv{^2 \vec A}{t^2} = - \mu_0 \vec J\]Back in the static case, this reduces to Poisson’s equation
\[\laplacian V = - \frac{1}{\epsilon_0} \rho\] \[\laplacian \vec A = - \mu_0 \vec J\]and we know how to solve these
\[V(\vec r) = \frac{1}{4 \pi \epsilon_0} \int \frac{\rho(\vec r')}{\gr} \dd \tau' \] \[\vec A(r) = \frac{\mu_0}{4 \pi} \int \frac{\vec J (\vec r')}{\gr} \dd \tau' \]We know that electromagnetic disturbances travel at the speed of light (at least in vacuum). So for general distributions of sources that may be changing in time, it’s not what the source is doing right now that matters - it’s what was happening at some earlier time (called the retarded time) when the “message” left. The information has traveled a distance \( \gr \), so the delay is \( \gr / c \), so the retarded time is
\[t_r \equiv t - \frac{\gr }{c}\]So we can immediately generalize our solutions for the potentials of static sources to the retarded potentials
\[V(\vec r, t) = \frac{1}{4 \pi \epsilon_0} \int \frac{\rho(\vec r', t_r)}{\gr} \dd \tau'\] \[\vec A(\vec r, t) = \frac{\mu_0}{4 \pi} \int \frac{\vec J(\vec r ', t_r)}{\gr} \dd \tau '\]That wasn’t so bad! It can be shown that the retarded potentials satisfy the inhomogeneous wave equations and the Lorenz gauge condition, giving some much-needed credibility to our argument that EM “messages” travel at the speed of light. While the math involved might be quite nasty (remember that \( \gr \) depends on \( | \vec r - \vec r’ | \)), in principle it’s straightforward to determine the fields directly by
\[\vec E = - \grad V - \pdv{\vec A}{t} \qquad \vec B = \curl \vec A\]Together, these are called the Jefimenko’s equations.