Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

In electrostatics the field followed from a scalar potential, $E = - \nabla V$ (Section 3.3), and in magnetostatics from a vector potential, $B = \nabla \times A$ (Section 4.1). Potentials are often easier to compute than fields, because they are integrals of the sources without the vector products of Coulomb's and Biot–Savart's laws. Do they survive when the fields vary in time?

The vector potential. $B = \nabla \times A$ still works. It was possible in magnetostatics because $\nabla \cdot B = 0$ (a field without divergence is the curl of some vector field), and the fourth Maxwell equation says that $\nabla \cdot B = 0$ holds for all fields, static or not. So we keep

B = \nabla \times A . (5.5.34)

The scalar potential. $E = - \nabla V$ does not survive, because it would give $\nabla \times E = 0$ and contradict Faraday's law. Substitute Eq. (5.5.34) into the first Maxwell equation instead:

\nabla \times E = - \frac{\partial B}{\partial t} = - \frac{\partial}{\partial t} \nabla \times A = - \nabla \times \frac{\partial A}{\partial t} . (5.5.35)

Collecting both terms under one curl,

\nabla \times (E + \frac{\partial A}{\partial t}) = 0 ⟹ E + \frac{\partial A}{\partial t} = - \nabla V, (5.5.36)

because a field with zero curl is the gradient of a scalar (Section 3.3), and we call that scalar $- V$ as in electrostatics. Finally we get

The fields from the potentials

E = - \nabla V - \frac{\partial A}{\partial t}, B = \nabla \times A . (5.5.37)

With these, $\nabla \cdot B = 0$ and Faraday's law hold identically, for any $V$ and $A$ .

In plain words

The electric field has two parts. $- \nabla V$ is the field of the charges, as in electrostatics. $- \partial A / \partial t$ is the induced field of Faraday's law: a changing current changes $A$ , and the rate of change of $A$ is an electric field with a curl.

The new term is why $V$ alone no longer describes $E$ , and why the "voltage" between two points depends on the path in a time-varying field (Section 5.3, Exercise 5.5.7). The potentials pay off in three ways (Figure 5.5.7): two of the four Maxwell equations are satisfied automatically, four unknown functions ( $V$ , $A_{x}$ , $A_{y}$ , $A_{z}$ ) replace the six components of $E$ and $B$ , and, with a suitable choice made below, the remaining two equations decouple.

**Figure 5.5.7.** The route through the potentials. The potentials are found from the sources by solving two wave equations (or by the retarded integrals of Section 5.5.10), and the fields then follow by differentiation.

The equation for A, step by step

Substitute Eq. (5.5.37) into the second Maxwell equation, written for a nonconducting medium ( $σ = 0$ , so that $J$ is the source current alone) as $\nabla \times B / μ = J + ε \partial E / \partial t$ :

\frac{1}{μ} \nabla \times \nabla \times A = J + ε \frac{\partial}{\partial t} (- \nabla V - \frac{\partial A}{\partial t}) . (5.5.38)

Multiply by $μ$ and expand the double curl with Eq. (5.5.6):

\nabla (\nabla \cdot A) - \nabla^{2} A = μ J - μ ε \nabla \frac{\partial V}{\partial t} - μ ε \frac{\partial ^{2} A}{\partial t ^{2}} . (5.5.39)

Collect the two gradients on one side:

\nabla^{2} A - μ ε \frac{\partial ^{2} A}{\partial t ^{2}} = - μ J + \nabla (\nabla \cdot A + μ ε \frac{\partial V}{\partial t}) . (5.5.40)

The definition $B = \nabla \times A$ fixes only the curl of $A$ . Its divergence is still ours to choose (Section 5.5.9 makes this precise), and the natural choice is the one that removes the bracket.

Definition · Lorenz gauge

The Lorenz condition, or Lorenz gauge for the potentials, is the choice

\nabla \cdot A + μ ε \frac{\partial V}{\partial t} = 0. (5.5.41)

With Eq. (5.5.41) the bracket in Eq. (5.5.40) vanishes and $A$ obeys a wave equation of its own, driven by the current alone:

\nabla^{2} A - μ ε \frac{\partial ^{2} A}{\partial t ^{2}} = - μ J . (5.5.42)

The equation for V

Substitute Eq. (5.5.37) into the third Maxwell equation, $\nabla \cdot E = ρ_{v} / ε$ :

\nabla \cdot (- \nabla V - \frac{\partial A}{\partial t}) = - \nabla^{2} V - \frac{\partial}{\partial t} (\nabla \cdot A) = \frac{ρ _{v}}{ε} . (5.5.43)

The Lorenz gauge replaces $\nabla \cdot A$ by $- μ ε \partial V / \partial t$ , and the scalar potential obeys a wave equation driven by the charge alone.

Wave equations for the potentials (Lorenz gauge)

\nabla^{2} A - μ ε \frac{\partial ^{2} A}{\partial t ^{2}} \nabla^{2} V - μ ε \frac{\partial ^{2} V}{\partial t ^{2}} \nabla \cdot A + μ ε \frac{\partial V}{\partial t} = - μ J, = - \frac{ρ _{v}}{ε}, = 0. (5.5.44)

In plain words

Both potentials obey the same wave equation, $V$ driven by $ρ_{v}$ and each Cartesian component of $A$ by the same component of $J$ . One method of solution serves all four.

Table 5.5.2 sets these results beside the static ones of Chapters 3 and 4. Each static equation gains one term, $- μ ε \partial^{2} / \partial t^{2}$ , and the gauge condition gains $μ ε \partial V / \partial t$ . Dropping these terms again is the quasi-static approximation of circuit theory, justified when the region of interest is small compared with the wavelength (Section 5.5.10).

Table 5.5.2. Potentials of static and of time-varying fields. The static equations are those of Sections 3.3 and 4.1 with $ε_{0}$ , $μ_{0}$ replaced by $ε$ , $μ$ .

	Static	Time-varying (Lorenz gauge)
gauge	$\nabla \cdot A = 0$	$\nabla \cdot A + μ ε \partial V / \partial t = 0$
scalar potential	$\nabla^{2} V = - ρ_{v} / ε$	$\nabla^{2} V - μ ε \partial^{2} V / \partial t^{2} = - ρ_{v} / ε$
vector potential	$\nabla^{2} A = - μ J$	$\nabla^{2} A - μ ε \partial^{2} A / \partial t^{2} = - μ J$
electric field	$E = - \nabla V$	$E = - \nabla V - \partial A / \partial t$
magnetic field	$B = \nabla \times A$	$B = \nabla \times A$

Lorenz, not Lorentz

The condition Eq. (5.5.41) is due to the Danish physicist L. V. Lorenz (1867). It is often printed as "Lorentz gauge", after H. A. Lorentz of the Lorentz force, who used it later. Cheng's book calls it the Lorentz condition.

5.5.8 Potential functions

The equation for A, step by step

The equation for V