Maxwell’s Equations (Integral Form)
It is sometimes easier to understand Maxwell’s equations in their integral form; the version we outlined last time is the differential form.
We pick any region we want and integrate both sides of each equation over that region:
On the left-hand sides we can use the divergence theorem, while the right sides can simply be evaluated:
where is the total charge contained within the region . Gauss’ law tells us that the flux of the electric field out through a closed surface is (basically) equal to the charge contained inside the surface, while Gauss’ law for magnetism tells us that there is no such thing as a magnetic charge.
Faraday’s law was basically given to us in integral form, but we can get it back from the differential form:
We pick any surface and integrate the flux of both sides through it:
On the left we can use Stokes’ theorem, while on the right we can pull the derivative outside the integral:
where is the flux of the magnetic field through the surface . Faraday’s law tells us that a changing magnetic field induces a current around a circuit.
A similar analysis helps with Ampère’s law:
We pick a surface and integrate:
Then we simplify each side.
where is the flux of the electric field through the surface , and is the total current flowing through the surface . Ampère’s law tells us that a flowing current induces a magnetic field around the current, and Maxwell’s correction tells us that a changing electric field behaves just like a current made of moving charges.
We collect these together into the integral form of Maxwell’s equations: