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The Wave Equation

The two "Laws" of ELECTRODYNAMICS - FARADAY'S LAW and AMPÈRE'S LAW - can be combined to produce a very important result.

First let's simplify matters by considering the behaviour of electromagnetic fields in empty space, where

$$\rho = 0, \qquad \Vec{J} = 0, \qquad \Vec{D} = \epsilon_\ci . . . . . . \quad \hbox{\rm and} \quad \Vec{B} = \mu_\circ \Vec{H} .$$

Our two equations then read

$$\Curl{E} = - {\partial \Vec{B} \over \partial t} \quad \h . . . . . . irc} = \epsilon_\circ {\partial \Vec{E} \over \partial t} .$$

We can simplify further by assuming that the electric field is in the $\hat{y}$ direction and the magnetic field is in the $\hat{z}$ direction. In that case,

$${\partial E \over \partial x} = - {\partial B \over \partial . . . . . . = - \epsilon_\circ \mu_\circ {\partial E \over \partial t}$$

where the second equation has been multiplied through by $\mu_\circ$.

If we now take the derivative of the first equation with respect to x and derivative of the second equation with respect to t, we get

$${\partial^2 E \over \partial x^2} = - {\partial^2 B \over \ . . . . . . \epsilon_\circ \mu_\circ {\partial^2 E \over \partial^2 t} .$$

$$\hbox{\rm Since} \qquad {\partial^2 B \over \partial x \partial t} = {\partial^2 B \over \partial t \partial x} ,$$

the combination of these two equations yields

$${\partial^2 E \over \partial x^2} = \epsilon_\circ \mu_\circ {\partial^2 E \over \partial^2 t}$$

which the discerning reader will recognize as the one-dimensional WAVE EQUATION for E,

$$\fbox{\hbox{$\displaystyle {\partial^2 E \over \partial x^2 . . . . . . re} \qquad c = {1 \over \sqrt{\epsilon_\circ \mu_\circ} } $}}$$

is the propagation velocity. You can easily show that there is an identical equation for B. A more general derivation yields the 3-dimensional version,

$$\Delsq{\Vec{E}} \; = \; {1 \over c^2} \; {\partial^2 \Vec{E . . . . . . partial t^2} \qquad \hbox{\rm or} \qquad \Box^2 \Vec{E} = 0$$

In either form, this equation expresses the fact that, since a changing electric field generates a magnetic field but that change in the magnetic field generates, in turn, an electric field, and so on, we can conclude that electromagnetic fields will propagate spontaneously through regions of total vacuum in the form of a wave of $\Vec{E}$ and $\Vec{B}$ fields working with/against each other.

This startling conclusion (in 1865) led to the revision of all the "classical" paradigms of Physics, even such fundamental concepts as space and time.