9.1: The Retarded Potentials

Nonhomogeneous Wave Equations

Maxwell's equations in complete generality are

\begin{align} \nabla \times \textbf{E}&=-\frac{\partial \textbf{B}}{\partial t}\\
\nabla \times \textbf{H}&=\textbf{J}_{f}+\frac{\partial \textbf{D}}{\partial t}\\
\nabla \cdot \textbf{B}&=0\\
\nabla \cdot \textbf{D}&=\rho _{f}\end{align}

In our development we will use the following vector identities

\[ \begin{align} 
\nabla \times \left ( \nabla V \right )&=0\\
\nabla \cdot \left (\nabla\times \textbf{A}\right )&=0\\
\nabla \times \left (\nabla\times \textbf{A}\right )&=\nabla\left (\nabla\cdot \textbf{A}\right )-\nabla^{2}\textbf{A}
\end{align} \]

where \(\textbf{A}\) and \(V\) can be any functions but in particular will be the magnetic vector potential and electric scalar potential, respectively.

Because in (3) the magnetic field has no divergence, the identity in (6) allows us to again define the vector potential \(\textbf{A}\) as we had for quasi-statics in Section 5-4:

\[ \textbf{B}=\nabla\times \textbf{A} \]

so that Faraday's law in (1) can be rewritten as

\[ \nabla\times \left ( \textbf{E}+\frac{\partial \textbf{A}}{\partial t} \right )=0 \]

Then (5) tells us that any curl-free vector can be written as the gradient of a scalar so that (9) becomes

\[ \textbf{E}+\frac{\partial \textbf{A}}{\partial t}=-\nabla V \]

where we introduce the negative sign on the right-hand side so that \(V\) becomes the electric potential in a static situation when \(\textbf{A}\) is independent of time. We solve (10) for the electric field and with (8) rewrite (2) for linear dielectric media \(\left ( \textbf{D}=\varepsilon \textbf{E},\textbf{B}=\mu \textbf{H} \right )\):

\[ \nabla\times \left ( \nabla\times \textbf{A} \right )=\mu \textbf{J}_{f}+\frac{1}{c^{2}}\left [ -\nabla\left ( \frac{\partial V}{\partial t} \right )-\frac{\partial^2 \textbf{A}}{\partial t^2} \right ],\quad c^{2}=\frac{1}{\varepsilon \mu } \]

The vector identity of (7) allows us to reduce (11) to

\[\nabla^{2}\textbf{A}-\nabla\left [ \nabla\cdot \textbf{A}+\frac{1}{c^{2}}\frac{\partial V}{\partial t} \right ]-\frac{1}{c^{2}}\frac{\partial^2 \textbf{A}}{\partial t^2}=-\mu \textbf{J}_{f} \]

Thus far, we have only specified the curl of \(\textbf{A}\) in (8). The Helmholtz theorem discussed in Section 5-4-1 told us that to uniquely specify the vector potential we must also specify the divergence of \(\textbf{A}\). This is called setting the gauge. Examining (12) we see that if we set

\[ \nabla\cdot \textbf{A}=-\frac{1}{c^{2}}\frac{\partial V}{\partial t} \]

the middle term on the left-hand side of (12) becomes zero so that the resulting relation between \(\textbf{A}\) and \(\textbf{J}_{f}\) is the nonhomogeneous vector wave equation:

\[ \nabla^{2}\textbf{A}-\frac{1}{c^{2}}\frac{\partial^2 \textbf{A}}{\partial t^2}=-\mu \textbf{J}_{f} \]

The condition of (13) is called the Lorentz gauge. Note that for static conditions, \( \nabla\cdot \textbf{A}=0\), which is the value also picked in Section 5-4-2 for the magneto-quasi-static field. With (14) we can solve for \(\textbf{A}\) when the current distribution \(\textbf{J}_{f}\) is given and then use (13) to solve for \(V\). The scalar potential can also be found directly by using (10) in Gauss's law of (4) as

\[ \nabla^{2}V+\frac{\partial }{\partial t}\left ( \nabla\cdot \textbf{A} \right )=\frac{-\rho _{f}}{\varepsilon } \]

The second term can be put in terms of \(V\) by using the Lorentz gauge condition of (13) to yield the scalar wave equation:

\[ \nabla^{2}V-\frac{1}{c^{2}}\frac{\partial V}{\partial t}=\frac{-\rho _{f}}{\varepsilon } \]

Note again that for static situations this relation reduces to Poisson's equation, the governing equation for the quasi-static electric potential.

Solutions to the Wave Equation

We see that the three scalar equations of (14) (one equation for each vector component) and that of (16) are in the same form. If we can thus find the general solution to any one of these equations, we know the general solution to all of them.

As we had earlier proceeded for quasi-static fields, we will find the solution to (16) for a point charge source. Then the solution for any charge distribution is obtained using superposition by integrating the solution for a point charge over all incremental charge elements.

In particular, consider a stationary point charge at \(r =0\) that is an arbitrary function of time \(\mathcal{Q}\left ( t \right )\). By symmetry, the resulting potential can only be a function of \(r\) so that (16) becomes

\[ \frac{1}{r^{2}}\frac{\partial }{\partial r}\left ( r^{2}\frac{\partial V}{\partial r} \right )-\frac{1}{c^{2}}\frac{\partial^2 V}{\partial t^2}=0,\quad r> 0 \]

where the right-hand side is zero because the charge density is zero everywhere except at \(r=0\). By multiplying (17) through by \(r\) and realizing that

\[ \frac{1}{r^{2}}\frac{\partial }{\partial r}\left ( r^{2}\frac{\partial V}{\partial r} \right )=\frac{\partial^2}{\partial r^2}\left ( rV \right ) \]

we rewrite (17) as a homogeneous wave equation in the variable \(\left ( rV \right )\):

\[ \frac{\partial^2}{\partial r^2}\left ( rV \right ) -\frac{1}{c^{2}}\frac{\partial^2 }{\partial t^2}\left ( rV \right )=0 \]

which we know from Section 7-3-2 has solutions

\[ rV=f_{+}\left ( t-\frac{r}{c} \right )+f_{-}\left ( t\underset{\nearrow \quad }{\overset{\qquad 0}{\overset{\quad \nearrow}{+}}} \frac{r}{c} \right ) \]

We throw out the negatively traveling wave solution as there are no sources for \(r>0\) so that all waves emanate radially outward from the point charge at \(r=0\). The arbitrary function \(f_{+}\) is evaluated by realizing that as \(r\rightarrow 0\) there can be no propagation delay effects so that the potential should approach the quasi-static Coulomb potential of a point charge:

\[ \lim_{r\rightarrow 0}V=\frac{\mathcal{Q}\left ( t \right )}{4\pi \varepsilon r}\Rightarrow f_{+}\left ( t \right )=\frac{\mathcal{Q}\left ( t \right )}{4\pi \varepsilon} \]

The potential due to a point charge is then obtained from (20) and (21) replacing time \(t\) with the retarded time \(t-r/c\):

\[ V\left ( r,t \right )=\frac{\mathcal{Q}\left ( t-r/c \right )}{4\pi \varepsilon r} \]

The potential at time \(t\) depends not on the present value of charge but on the charge value a propagation time \(r/c\) earlier when the wave now received was launched.

The potential due to an arbitrary volume distribution of charge \(\rho _{f}\left ( t \right )\) is obtained by replacing \(\mathcal{Q}\left ( t-r/c \right )\) with the differential charge element \(\rho _{f}\left ( t \right )d\textrm{V}\) and integrating over the volume of charge:

\[ V\left ( r,t \right )=\int_{\textrm{all charge}}\frac{\rho _{f}\left ( t-r_{\mathcal{Q}P}/c \right )}{4\pi \varepsilon r_{\mathcal{Q}P}}d\textrm{V} \]

where \(r_{\mathcal{Q}P}\) is the distance between the charge as a source at point \(\mathcal{Q}\) and the field point at \(P\).

The vector potential in (14) is in the same direction as the current density \(\textbf{J}_{f}\). The solution for \(\textbf{A}\) can be directly obtained from (23) realizing that each component of \(\textbf{A}\) obeys the same equation as (16) if we replace \(\rho _{f}/\varepsilon \) by \(\mu \textbf{J}_{f}\):

\[ A\left ( r,t \right )=\int_{\textrm{all current}}\frac{\mu \textbf{J}_{f}\left ( t-r_{\mathcal{Q}P}/c \right )}{4\pi \varepsilon r_{\mathcal{Q}P}}d\textrm{V} \]