Introduction to Electrodynamics/Chapter 8/1/1

Where there are no charges or currents, we can simplify Maxwell's Equations to this basic form:

<math> \begin{align} \nabla\cdot\vec{E} &=0 \\ \nabla\cdot\vec{B} &=0 \\ \nabla\times\vec{E} &=-{\partial \vec{B} \over \partial t} \\ \nabla\times\vec{B} &=\mu_0\epsilon_0{\partial \vec{E} \over \partial t} \\ \end{align} </math>

These equations are coupled: E depends on B and vice-versa. We can decouple them taking the curl of the curl equations.

<math> \begin{align}

\nabla\times(\nabla\times\vec{E}) &=\nabla\times\left(-{\partial \vec{B} \over \partial t}\right) \\ \nabla(\nabla\cdot\vec{E}) - \nabla^2\vec{E} &=-{\partial \over \partial t}\nabla\times \vec{B} \\ - \nabla^2\vec{E} &=-{\partial \over \partial t}\mu_0\epsilon_0{\partial \vec{E} \over \partial t} \\ \nabla^2\vec{E} &= \mu_0\epsilon_0{\partial^2 \vec{E} \over \partial t^2} \\

\end{align} </math>

• Recall the curl of a curl is the gradient of the divergence minus the laplacian.
• The curl of a time derivative is the time derivative of a curl since the curl doesn't have any relation to time.
• Substitute in Maxwell's Correction.
• There is no electric charge, so the divergence of E is zero.
• Both sides end up being negative.

(I won't do the same for the B field. I'll leave that as a trivial exercise for you.)

Now you have 3 separate equations for E and 3 for B. They are decoupled, meaning E doesn't depend on B and vice-versa. But now they are second-order differential equations, which leads to a certain kind of result.

Second-order differential equations of the form:

<math> \nabla^2 f = {1 \over v^2}{\partial^2 f \over \partial t^2} </math>

are called wave equations. We've already seen them arise for a variety of motions in classical mechanics. (If you haven't, or you don't remember, now's a good time to review that material.)

The constant v is the velocity of the wave. We'll talk about that and other properties of waves in the next section.

For now, notice that:

<math> \begin{align} {1 \over v^2} &= \mu_0\epsilon_0 \\ v &= \sqrtTemplate:1 \over \mu 0\epsilon 0 \\ v &= \sqrt{{1 \over (4\pi\times 10^{-7} N/A^2) (8.85 \times 10^{-12} C^2/Nm^2)}} \\ v &= \sqrt{{1 \over 1.11 \times 10^{-17} C^2/A^2m^2}} \\ v &= \sqrt{9.01 \times 10^{16} m^2/s^2} \\ v &= 3.00 \times 10^{8} m/s \\ v &= c \\

\end{align} </math>

I hope this isn't new for you. It should be awe-inspiring, however. These two constants, combined in this way, give you the speed of light. This says that light is really just electromagnetic waves. They are one and the same.

Now we have united light and matter through electromagnetic forces. Think about this and allow yourself to be awed by this result.

Let's look carefully at what we've done. We've started with Coulomb's law which gives us a constant epsilon-naught which is the relationship between force, distance, and charge.

We've also applied Maxwell's correction to Ampere's law, which relates changes in the electric field over time to the curl of a magnetic field. Ampere's Law gives us mu-naught, which related currents, distance, and force.

Then we brought them together by a simple substitution to derive a wave equation for empty space whose velocity exactly matches the speed of light we've measured with other experiments.

To those who discovered this fact, that light is really an electromagnetic wave, this was a shocking and surprising discovery. To us today, it's hardly surprising.