Differential Forms and Maxwell’s Equations
Contents
22. Differential Forms and Maxwell’s Equations#
22.1. Gauss’s Law in Differential Form#
Lets revisit what we have learnt about Gauss’s law for electric charges
since the charge
If we recall the divergence theorem, it states that the flux of a vector field passing a closed surface area is related to the divergence of the vector field within the volume, so for an electric field
so we can write that, up to a boundary term
in order for this expression to be true for any volume
Likewise we can do the same for magnetic fields, which in the absence of magnetic monopoles will look like:
22.2. Ampère’s Law in Differential Form#
We can revisit Amp`ere’s law, with the displacement current:
Given we know the Stoke’s theorem tells us that the summing of a vector field around a closed loop will be related to the curl of the field, for the magnetic field:
Also we can think about the current
which all put together gives:
in order for this expression to be true for any bounding area
22.3. Faraday’s Law in Differential Form#
Finally looking at Faraday’s law again:
The EMF
where the final sign on the integral is given by the convention for clockwise conventional current. Fig. 21.1 shows that a positive
change in
We can rewrite this using Stoke’s theorem:
and so this can all be written as:
in order for this expression to be true for any bounding volume
22.4. Maxwell’s Equations in Vacuum#
We can collect together Equations (22.1), (22.2), (22.3) and (22.4) and present as Maxwell’s equations:
These are local equations, they can be solved at points in space
22.5. Maxwell’s Equations in Matter*#
Recall that all the Maxwell equations in (22.5) are only valid in vacuum, within matter however electromagnetic fields are better described in terms of the
where
22.6. Magnetic Monopoles*#
If magnetic monopoles were found to exist, then we could add the additional terms to Maxwell’s equations that would need to be present:
where
Additionally if there are
where