15.12: Retarded Potential

In a static situation, in which the charge density \(\rho\), the current density \(\textbf{J}\), the electric field \(\textbf{E}\) and potential \(V\), and the magnetic field \(\textbf{B}\) and potential \(\textbf{A}\) are all constant in time (i.e. they are functions of \(x\), \(y\) and \(z\), but not of \(t\)) we already know how to calculate, in vacuo, the electric potential from the electric charge density and the magnetic potential from the current density. The formulas are

\[V(x,y,z) = \dfrac{1}{4 \pi \epsilon_o} \int \dfrac{\rho(x^{\prime},y^{\prime},z^{\prime}) dv^{\prime}}{R} \tag{15.12.1} \label{15.12.1}\]

and

\[\textbf{A}(x,y,z) = \dfrac{\mu_o}{4 \pi } \int \dfrac{\textbf{J} (x^{\prime},y^{\prime},z^{\prime}) dv^{\prime}}{R} \tag{15.12.2} \label{15.12.2}\]

Here \(R\) is the distance between the point \((x^{\prime},y^{\prime},z^{\prime})\) and the point \((x,y,z)\) and \(v^{\prime}\) is a volume element at the point \((x^{\prime},y^{\prime},z^{\prime})\). I can’t remember if we have written these two equations in exactly that form before, but we have certainly used them, and given lots of examples of calculating \(V\) in Chapter 2, and one of calculating \(\textbf{A}\) in Section 9.3.

The question we are now going to address is whether these formulas are still valid in a nonstatic situation, in which the charge density \(\rho\), the current density \(\textbf{J}\), the electric field \(\textbf{E}\)and potential \(V\), and the magnetic field \(\textbf{B}\) and potential \(\textbf{A}\) are all varying in time (i.e. they are functions of x, y, z and t). The answer is “yes, but…”. The relevant formulas are indeed

\[V(x,y,z,t) = \dfrac{1}{4 \pi \epsilon_o} \int \dfrac{\rho(x^{\prime},y^{\prime},z^{\prime},t^{\prime}) dv^{\prime}}{R} \tag{15.12.3} \label{15.12.3}\]

and

\[\textbf{A}(x,y,z,t) = \dfrac{\mu_o}{4 \pi } \int \dfrac{\textbf{J} (x^{\prime},y^{\prime},z^{\prime},t^{\prime}) dv^{\prime}}{R} \tag{15.12.4} \label{15.12.4}\]

…but notice the \(t^{\prime}\) on the right hand side and the \(t\) on the left hand side! What this means is that, if \(\rho(x^{\prime},y^{\prime},z^{\prime},t^{\prime})\) is the charge density at a point \((x^{\prime},y^{\prime},z^{\prime})\) at time \(t^{\prime}\), equation \(\ref{15.12.3}\) gives the correct potential at the point \((x^{\prime},y^{\prime},z^{\prime})\) at some slightly later time \(t\), the time difference \(t-t^{\prime}\) being equal to the time \(R/c\) that it takes for an electromagnetic signal to travel from \((x,y,z)\) to \((x^{\prime},y^{\prime},z^{\prime})\). If the charge density at \((x^{\prime},y^{\prime},z^{\prime})\) changes, the information about this change cannot reach the point instantaneously; it takes a time \(R/c\) for the information to be transmitted from one point to another. The same considerations apply to the change in the magnetic potential when the current density changes, as described by equation \(\ref{15.12.4}\). The potentials so calculated are called, naturally, the retarded potentials. While this result has been arrived at by a qualitative argument, in fact equations \(\ref{15.12.3}\) and \(\ref{15.12.4}\) can be obtained as a solution of the differential Equations 15.11.12 and 15.11.13. Mathematically there is also a solution that gives an “advance potential” – that is, one in which \(t^{\prime}-t\) rather than \(t-t^{\prime}\) is equal to \(R/c\). You can regard, if you wish, the retarded solution as the “physically acceptable” solution and discard the “advance” solution as not being physically significant.That is, the potential cannot predict in advance that the charge density is about to change, and so change its value before the charge density does. Alternatively one can think that the laws of physics, from the mathematical view at least, allow the universe to run equally well backward as well as forward, though in fact the arrow of time is such that cause must precede effect (a condition which, in relativity, leads to the conclusion that information cannot be transmitted from one place to another at a speed faster than the speed of light). One is also reminded that the laws of physics, from the mathematical view at least, allow the entropy of an isolated thermodynamical system to decrease (see Section 7.4 in the Thermodynamics part of these notes) – although in the real universe the arrow of time is such that the entropy in fact increases. Recall also the following passage from Through the Looking-glass and What Alice Found There.

Addendum. Coincidentally, just two days after having completed this chapter, I received the 2005 February issue of Astronomy & Geophysics, which included a fascinating article on the Arrow of Time. You might want to look it up. The reference is Davis, P., Astronomy & Geophysics (Royal Astronomical Society) 46, 26 (2005).