Maxwell's Equations

We are now in a position to write out a complete, mathematically self-consistent, set of field equations that govern electric and magnetic phenomena:

$$\nabla\cdot{\bf E} = \frac{\rho}{\epsilon_0},$$ (2.484)

$$\nabla\cdot{\bf B} =0,$$ (2.485)

$$\nabla\times{\bf E} = - \frac{\partial {\bf B}}{\partial t} ,$$ (2.486)

$$\nabla\times{\bf B} = \mu_0\, {\bf j}+ \epsilon_0\,\mu_0\, \frac{\partial {\bf E}}{\partial t}.$$ (2.487)

The equations are known as Maxwell's equation. The first Maxwell equation describes how electric fields are induced by electric charges, and is equivalent to Gauss's law. The second Maxwell equation states that there is no such thing as a magnetic monopole. The third Maxwell equation describes the induction of electric fields by changing magnetic fields, and is equivalent to Faraday's law of electromagnetic induction. The fourth Maxwell equation describes the generation of magnetic fields by electric currents, and the induction of magnetic fields by changing electric fields, and incorporates Ampère's circuital law.

As an example of a calculation involving the displacement current, let us find the current and displacement current densities associated with the decaying charge distribution

$$\rho(r,t) = \frac{\rho_0\,\exp(-t/\tau)}{r^2+a^2},$$ (2.488)

where $r$ is a spherical polar coordinate (see Section A.23), $\rho_0$ is a constant, and $\tau$ and $a$ are positive constants. Now, according to charge conservation,

$$\nabla\cdot{\bf j} = -\frac{\partial \rho}{\partial t}.$$ (2.489)

[See Equation (2.473).] By symmetry, we expect ${\bf j}={\bf j}(r,t)$. Hence, it follows that ${\bf j} = j_r(r,t)\,{\bf e}_r$ [because only a radial current has a non-zero divergence when ${\bf j}= {\bf j}(r)$]. [See Equation (A.173).] Thus, the previous two equations yield.

$$\frac{1}{r^2}\,\frac{\partial (r^2\,j_r)}{\partial r} = - \frac{\partial \rho}{\partial t} = \frac{\rho_0\,\exp(-t/\tau)}{\tau\,(r^2+a^2)}.$$ (2.490)

The previous expression can be integrated, subject to the sensible boundary condition $j_r(0)= 0$, to give

$$j_r(r) = \frac{\rho_0}{\tau}\,{\rm e}^{-t/\tau}\left[\frac{r-a\,\tan^{-1}(r/a)}{r^2}\right].$$ (2.491)

Now, the electric field generated by the decaying charge distribution satisfies

$$\nabla \cdot {\bf E} = \frac{\rho}{\epsilon_0}.$$ (2.492)

[See Equation (2.466).] Because $\partial\rho/\partial t=-\rho/\tau$, it can be seen, from a comparison of Equations (2.489) and (2.492), that

$${\bf E} =\frac{\tau}{\epsilon_0}\,{\bf j}.$$ (2.493)

However, the displacement current density is given by

$${\bf j}_d = \epsilon_0\,\frac{\partial {\bf E}}{\partial t} = - {\bf j},$$ (2.494)

because $\partial {\bf j}/\partial t = - {\bf j}/\tau$. Hence, we conclude that the displacement current density cancels out the true current density, so that ${\bf j} + {\bf j}_d = {\bf0}$. This is just as well, because $\nabla\times {\bf B} = \mu_0\,({\bf j} + {\bf j}_d)$. [See Equation (2.479).] But, if ${\bf B}= {\bf B}(r,t)$ then, by symmetry, $\nabla\times {\bf B}$ has no radial component. [See Equation (A.174).] Thus, if the current and displacement current are constrained, by symmetry, to be radial, then they must sum to zero, otherwise the fourth Maxwell equation cannot be satisfied. In fact, no magnetic field is generated in this particular example.