Retarded potentials

Solution of Maxwell equations can be obtained from solution of wave equations.
Here are the wave equations for vector and scalar potentials:
\[ \nabla^2 \vec{A} - \frac{1}{c^2} \frac{\partial^2 \vec{A} }{\partial t^2} = -\mu_0 \cdot \vec{J} \tag{i}\] \[ \nabla^2 \Phi - \frac{1}{c^2} \frac{\partial^2 \Phi }{\partial t^2} = - \frac{\rho}{\epsilon_0} \tag{ii}\]
What can we notice looking at these equations?

Equations are linear. It means that if \( \vec{A}_1(\vec{r},t ) \) is the solution for \( \vec{J}_1\) and \( \vec{A}_2(\vec{r},t ) \) is the solution for \( \vec{J}_2\), then \( \vec{A}_1(\vec{r},t )+ \vec{A}_2(\vec{r},t ) \) satisfies the equation for \( \vec{J}_1+\vec{J}_2\).
This is handy because we can try to find solution for simple \( \rho(x,y,z,t) \) and then build source we want adding together many simple sources.

Solutions for Maxwell equations with field sources.

For Maxwell equations in free-space () we found two solutions:
  1. Plane wave: \[ \Phi(x,y,z,t) = f(t - r/c) \]
  2. Spherical wave: \[ \Phi(x,y,z,t) = f(t - r/c) / r \]
where \( f(t - r/c) \) is any twice-differentiable function. Now lets modify these free-space solutions so they are also solutions of equations with sources. What limitations we should impose on \( f()\) so that it is a solution at the source point?
Interfacing free-space solutions with the source.

If \( r \rightarrow 0 \) (closer and closer to the source) then \( f(t - r/c) \rightarrow f(t)\): wave did not have time to fly away yet. Then out spherical wave solution is: \[ \Phi(x,y,z,t) \rightarrow f(t) / r \] It behaves ( dropping as \( 1/r \) ) like potential of localized charge.
Also as \( r \rightarrow 0 \) spatial derivatives ( \( \nabla^2 \Phi \) ) become really big . Time derivative stay same (it does not depend upon \(r\) ). If we are close enough to the source: \[ \nabla^2 (f(t) / r) - \frac{1}{c^2} \frac{\partial^2 (f(t) / r) }{\partial t^2} \rightarrow \nabla^2 (f(t) / r) \qquad \text{so} \qquad \nabla^2 \Phi \rightarrow - \frac{\rho}{\epsilon_0} \tag{iii}\]

For localized charge , changing with time, potential is: \[ V(x,y,z,t) = - \frac{Q(t)}{4 \pi \epsilon_0 r} \tag{iv} \] It satisfies Poisson's equation: \[ \nabla^2 V(x,y,z,t) = - \frac{Q(t)}{ \epsilon_0 r} \tag{v} \]

Equation (v) is similar to (iii), and if (iv) is solution for (v) then solution for (iii) is \[ f(t, r \rightarrow 0) = \frac{\rho(t)}{ 4 \pi} \] For any point in space \[ f(t, r) = \frac{\rho(t -r/c )}{ 4 \pi} \]

Retarded potentials

Solution happens to be surprisingly simple:
Potential \( \Phi(x,y,z,t) \) follows \( \rho(x,y,z,t) \) in time but delayed. If observation point is far away from source point then potential \( \Phi(x,y,z,t) \) is \( \rho(x,y,z,t-r/c) \) where \(r\) is distance between source and observation point .
Final solution is:
\[ \Phi() = \frac{1}{4 \pi \epsilon} \int_V {} dV \] \[ \nabla^2 \Phi - \frac{1}{c^2} \frac{\partial^2 \Phi }{\partial t^2} = - \frac{\rho}{\epsilon_0} \]
When calculating these integrals, retard.svg

Retarded Fields

Solution for Maxwell equations is spherical wave which propagates away from the source with speed of light.

Using Retarded Potentials: Exampls