The Unapologetic Mathematician

Mathematics for the interested outsider

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.

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February 1, 2012 - Posted by | Electromagnetism, Mathematical Physics

10 Comments »

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

    Pingback by Maxwell’s Equations (Integral Form) « The Unapologetic Mathematician | February 2, 2012 | Reply

  2. [...] to note at this point that we didn’t have to start with our experimentally-justified axioms. Maxwell’s equations suffice to derive all the physics we [...]

    Pingback by Deriving Physics from Maxwell’s Equations « The Unapologetic Mathematician | February 3, 2012 | Reply

  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. [...]

    Pingback by The Electromagnetic Wave Equations « The Unapologetic Mathematician | February 7, 2012 | Reply

  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 [...]

    Pingback by Plane Waves « The Unapologetic Mathematician | February 8, 2012 | Reply

  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 [...]

    Pingback by The Propagation Velocity of Electromagnetic Waves « The Unapologetic Mathematician | February 9, 2012 | Reply

  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 [...]

    Pingback by Polarization of Electromagnetic Waves « The Unapologetic Mathematician | February 10, 2012 | Reply

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

    Pingback by Maxwell’s Equations in Differential Forms « The Unapologetic Mathematician | February 22, 2012 | Reply

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

    Pingback by The Meaning of the Speed of Light « The Unapologetic Mathematician | February 24, 2012 | Reply

  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 [...]

    Pingback by The Faraday Field « The Unapologetic Mathematician | March 6, 2012 | Reply

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

    Pingback by The Higgs Mechanism part 1: Lagrangians « The Unapologetic Mathematician | July 16, 2012 | Reply


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