Physics with Calculus/Electromagnetism/Maxwell's Equations

So, to unify everything we have said about electricity and magnetism so far, we will write out Maxwell's equations together. Taken together, Maxwell's equations describe all of classical electrodynamics.

First we have Gauss' law:

${\oint }_S\overrightarrow{E}\cdot d\overrightarrow{A}=\frac{Q_{encl}}{{ϵ}_0}$.

Gauss' law for magnetism is

${\oint }_S\overrightarrow{B}\cdot d\overrightarrow{A}=0$.

Ampere's law is

${\oint }_C\overrightarrow{B}\cdot d\overrightarrow{s}={\mu }_0{I}_{encl}$.

Finally, Faraday's law is

${\oint }_C\overrightarrow{E}=-\frac{d{\mathrm{\Phi }}_B}{dt}$.

However, imagine a parallel plate capacitor. If we apply Ampere's law around the wire, and take the surface so that the wire cuts through it, we get one answer. But if we take the surface so that it goes in between the parallel plates, we get 0 for the surface integral! We have to revise our laws. Maxwell added in a term which essentially implies that when charge builds up, current has flowed in -- the conservation of charge. This manifests itself as a new term in Ampere's law (you can actually show that in order to have charge conserved, this is the only possible term that is legal and able to be measured):

${\oint }_C\overrightarrow{B}\cdot d\overrightarrow{s}={\mu }_0{I}_{encl}+{ϵ}_0{\mu }_0\frac{d{\mathrm{\Phi }}_E}{dt}$.

Look at the equations; there are two contour integral equations and two surface integral equations. The Ampere-Maxwell law and the Faraday law both have the derivative of the flux of the opposite field in them. It is not difficult to imagine that if we had a magnetic charge, Maxwell's equations would be completely symmetrical, up to the constants in front of terms.

Also, we can apply Green's Theorem to Maxwell's equations and obtain them in another form:

$\mathrm{\nabla }\cdot E=\frac{\rho }{{ϵ}_0}$

$\mathrm{\nabla }\cdot B=0$

$\mathrm{\nabla }\times E+\frac{\partial B}{\partial t}=0$

$\mathrm{\nabla }\times B-{ϵ}_0{\mu }_0\frac{\partial E}{\partial t}={\mu }_{0}J$.

Helmholtz's theorem states that a function is uniquely defined by it's divergence, curl, and suitable boundary conditions. If we play around with that theorem a little bit, we can show that Maxwell's equations completely specify all of electromagnetism.

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