# The Unapologetic Mathematician

## Maxwell’s Equations

Okay, let’s see where we are. There is such a thing as charge, and there is such a thing as current, which often — but not always — arises from charges moving around.

We will write our charge distribution as a function $\rho$ and our current distribution as a vector-valued function $J$, though these are not always “functions” in the usual sense. Often they will be “distributions” like the Dirac delta; we haven’t really gotten into their formal properties, but this shouldn’t cause us too much trouble since most of the time we’ll use them — like we’ve used the delta — to restrict integrals to smaller spaces.

Anyway, charge and current are “conserved”, in that they obey the conservation law:

$\displaystyle\nabla\cdot J=-\frac{\partial\rho}{\partial t}$

which states that the mount of current “flowing out of a point” is the rate at which the charge at that point is decreasing. This is justified by experiment.

Coulomb’s law says that electric charges give rise to an electric field. Given the charge distribution $\rho$ we have the differential contribution to the electric field at the point $r$:

$\displaystyle dE(r)=\frac{1}{4\pi\epsilon_0}\rho\frac{r}{\lvert r\rvert^3}dV$

and we get the whole electric field by integrating this over the charge distribution. This, again, is justified by experiment.

The Biot-Savart law says that electric currents give rise to a magnetic field. Given the current distribution $J$ we have the differential contribution to the magnetic field at the poinf $r$:

$\displaystyle dB(r)=\frac{\mu_0}{4\pi}J\times\frac{r}{\lvert r\rvert^3}dV$

which again we integrate over the current distribution to calculate the full magnetic field at $r$. This, again, is justified by experiment.

The electric and magnetic fields give rise to a force by the Lorentz force law. If a test particle of charge $q$ is moving at velocity $v$ through electric and magnetic fields $E$ and $B$, it feels a force of

$\displaystyle F=q(E+v\times B)$

But we don’t work explicitly with force as much as we do with the fields. We do have an analogue for work, though — electromotive force:

$\displaystyle\mathcal{E}=-\int\limits_CE\cdot dr$

One unexpected source of electromotive force comes from our fourth and final experimentally-justified axiom: Faraday’s law of induction

$\displaystyle\mathcal{E}=\frac{\partial}{\partial t}\int\limits_\Sigma B\cdot dS$

This says that the electromotive force around a circuit is equal to the rate of change of magnetic flux through any surface bounded by the circuit.

Using these four experimental results and definitions, we can derive Maxwell’s equations:

\displaystyle\begin{aligned}\nabla\cdot E&=\frac{1}{\epsilon_0}\rho\\\nabla\cdot B&=0\\\nabla\times E&=-\frac{\partial B}{\partial t}\\\nabla\times B&=\mu_0J+\epsilon_0\mu_0\frac{\partial E}{\partial t}\end{aligned}

The first is Gauss’ law and the second is Gauss’ law for magnetism. The third is directly equivalent to Faraday’s law of induction, while the last is Ampère’s law, with Maxwell’s correction.

February 1, 2012 -

1. [...] is sometimes easier to understand Maxwell’s equations in their integral form; the version we outlined last time is the differential [...]

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3. [...] Maxwell’s equations give us a collection of differential equations to describe the behavior of the electric and magnetic fields. Juggling them, we can come up with other differential equations that give us more insight into how these fields interact. And, in particular, we come up with a familiar equation that describes waves. [...]

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4. [...] derived a “wave equation” from Maxwell’s equations, but it’s not clear what it means, or even why this is called a wave equation. Let’s [...]

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5. [...] we’ve derived the wave equation from Maxwell’s equations, and we have worked out the plane-wave solutions. But there’s more to Maxwell’s [...]

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6. [...] look at another property of our plane wave solutions of Maxwell’s equations. Specifically, we’ll assume that the electric and magnetic fields are each plane waves in the [...]

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7. [...] again with Maxwell’s equations, we see all these divergences and curls which, though familiar to many, are really heavy-duty [...]

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8. [...] pick up where we left off last time converting Maxwell’s equations into differential [...]

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9. [...] factor to put time and space measurements on an equal footing, let’s actually do it to Maxwell’s equations. We start by moving the time derivatives over on the same side as all the space [...]

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10. [...] other two of Maxwell’s equations come automatically from taking the potentials as fundamental and coming up with the electric and [...]

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