# Classical Electromagnetism

## Chapter 3: Maxwell's Equations

### By Mark Lawrence

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## Electromagnetism in a nutshell

In about 1860, James Clerk Maxwell summarized and completed the knowledge of electromagnetism with his theory, Maxwell's Equations:

Ñ · E = ρ

Ñ x E = - dB / dt

Ñ · B = 0

Ñ x B = J - dE / dt

About 20 years later, Lorentz wrote down the force equation,

F = q(E + v x B)

Today we usually note that Maxwell's Equations can be simplified by using a scalar potential φ and a vector potential A,

E = -Ñφ - dΑ/dt

B = Ñ x A

resulting in

Ñ2φ  - 1 / c2 d2φ / dt2 = -ρ

Ñ2A  - 1 / c2 d2A/ dt2 = -J

Einstein recognized that these equations could be put into one equation by combining φ and A into a four vector, and ρ and J into another four vector, resulting in

Aμ,νν - Aν,μν = Jμ

That's it then, all of electromagnetism in one simple equation. Well, perhaps there's just a little more to discuss about this subject.

We'll start with electrostatics, the theory of electricity when nothing is changing in time. This means that all the time derivatives are zero, and we won't consider any magnetic fields. In this case, all we need is Poisson's equation,

Ñ2Φ  = -ρ

or if there are no charges present, LaPlace's equation

Ñ2Φ  = 0

It will turn out that these exceedingly simply equations have fantastically rich solutions. It will also turn out we will only be able to solve them in a few particular circumstances, circumstances which include enough symmetry and other simplifications so that we can make huge simplifying assumptions.

 dψ' = ih eiχ(x,t) ( dψ dχ ) = ih eiχ(x,t) ( d dχ ) dψ ih + ψ + dt dt dt dt dt

 -h2 d2ψ' -h2 eiχ(x,t)( d2ψ dψ dχ d2χ ) = -h2 eiχ(x,t) ( d dχ )2 ψ = + 2 + ψ + 2m dx2 2m dx2 dx dx dx2 2m dx dx