Maxwell's Theory Of Wave Propagation

Question

What is Maxwell's theory of Wave propagation and what is its physical interpretation?

Answer 1

Before Maxwell, the known laws of electromagnetism were thought to be:

$$\begin{array}{ll} \nabla \cdot \vec{D} = \rho& \text{Gauss's Law for Electricty}\\ \nabla \cdot \vec{B} = 0& \text{Gauss's Law for Magnetism}\\ \nabla \wedge \vec{E} = -\partial_t \vec{B}& \text{Faraday's Law}\\ \nabla \wedge \vec{H} = \vec{J}& \text{Amp}\grave{\text{e}}\text{re's Law}\\ \end{array}\tag{1}$$

where $\vec{E},\,\vec{H},\,\vec{D},\,\vec{B}$ are the electric and magnetic field, electric displacement and magnetic induction, respectively and $\rho,\,\vec{J}$ the electric charge and current density vectors, respectively.

The great problem was that Ampère's law as it stood was inconsistent with the principle of conservation of charge. If you take the vector divergence of both sides of the before-Maxwell Ampère law, you get $\nabla\cdot\vec{J} = 0$, whereas the general statement of charge conservation is $\nabla\cdot\vec{J} = -\partial_t \rho$. The latter form expresses the fact, for example, that charges can build up on capacitor plates as long as the nett charge of the system doesn't change, whereas $\nabla\cdot\vec{J} = 0$ means that current flow lines cannot end anywhere (as they do on capacitor plates in time-varying circuits). I say more about the intuitive physical interpretation of this discrepancy in my answer here. To cut a long story short, one way (not the only one) to get rid of these inconsistencies is to postulate that the RHS of Ampère's law should instead be $J+\partial_t \vec{D}$; then if you take the divergence of both sides of Ampère's law you get the continuity equation $\nabla\cdot\vec{J} = -\partial_t \rho$. This is exactly what Maxwell did.

Once this is done, it can be shown that all the six Cartesian components of the electric and magnetic field vectors in freespace fulfill the D'Alembert wave equation:

$$c^2 \nabla^2 \psi = \partial_t^2 \psi$$

which is the equation for a dispersionless wave. To see this, in one spatial dimension the equation is:

$$c^2 \partial_x^2 \psi = \partial_t^2 \psi$$

and its general solution is:

$$\psi = f(x-c\,t) + g(x + c\,t)$$

where $f$ and $g$ are arbitrary, continuously twice differentiable ($C^2$) functions. They are equations for pulses with invariant shape (defined by $f$ and $g$) running at speed $c$.