Maxwell Equations

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Maxwell’s Equations for Electromagnetic Waves

First, a word of caution. We’re going to work in CGS units, and some of the reference material above use SI units. In CGS units, there is no (it’s incorporated into the definition of charge) and becomes . In these units, Maxwell’s equations are written:

where is the electric field, is the magnetic field, is the current (charge per time), and is the charge density. In words, these equations state:

  1. Electrical charge creates a divergence of in the electric field (which gives rise to a radial potential).
  2. Magnetic fields do not diverge. There are no magnetic “charges" (that we know of).
  3. A magnetic field that changes with time creates a circular electric potential that drives charges around loops.
  4. A moving electrical charge (a “current") or an electric field that changes with time creates a circular magnetic potential.

1 Maxwell’s Equations in Free Space

In free-space, where (no charge) and (no current), Maxwell’s equations say

Using the vector identity

the curl () of the last 2 Maxwell’s equations reduces to the following wave equations (using that the divergence of each field is 0):

Why are these called wave equations? To make it easier to see why, let’s make them one dimensional.

1.1 A One-Dimensional Wave Equation

Along one dimension (let’s say, ), the wave equation for the electric field (derived above as a solution of Maxwell’s Equations in free space) is written

Let’s suppose that , and we define a new variable . The Chain Rule tells us that a spatial derivative of gives us

Similarly, for the time derivative, we have

Based on these identities, we have

If you plug these into the wave equation at the top of this section, you’ll see that it solves the equation. In the general 3D case for and , this means that any electromagnetic waveform that translates at speed solves these equations. And I probably needn’t point out that an electromagnetic waveform traveling at the speed of light is, well... light!

To see how this generalizes to plane waves, see Electromagnetic Plane Waves.

1.2 The Relative Orientation of and

From the previous section, we see that in free space, the Maxwell equations give us 2 wave equations. One propagates the -field at the speed of light, and the other propagates the -field at the speed of light, but we haven’t related the and fields to one another. To do this, we need to re-examine the Maxwell equations for free space.

We’ll defer a rigorous derivation of this until we’ve examined plane waves as a solution to these wave equations. In the meantime, here’s a sketch of the arguments for the relative phases of and . Let’s suppose that and are both described by sine waves of the same frequency, but with some arbitrary phase relative to one another. Let’s say that , so that at . Since a sine wave is anti-symmetric around 0, one can draw a loop encircling , and since is positive on one side of that loop, and negative on the other side, is not zero, and in fact, is maximal at .

CurlE.png


The curl of a sine wave (in 2 or more dimensions) is maximized around .

Since we know that

It must be true that is maximized at . For , this happens when . So by this hand-wavy argument (which we can can make more rigorous later), and must be in-phase with one another (i.e. they hit 0 at the same time). Moreover, because is perpendicular to and perpendicular to the spatial gradient of (which we’ve set to be in the direction, the direction the wave moves with time), must be perpendicular to and perpendicular to the direction the wave propagates. Finally, if you work out the spatial and temporal derivatives at , you’ll see that .