Maxwell's equations for the electromagnetic field [ in units with ] are:
Defining the anti- symmetric tensor with components:
the electric and magnetic fields are given by
If we also define a current four- vector :
Maxwell's equations can be written as [ Assignment 4 ]
where . We have now expressed Maxwell's equations in tensor form as required by Special Relativity.
The first of these equations implies charge conservaton
By performing a Lorentz transformation to a frame moving with speed v in the x direction, one can calculate how the electric and magnetic fields change:
We find [ Assignment 4 ] that is unchanged, while
where and is the electric field parallel and perpendicular to . Thus and get mixed.
The four- force on a particle of charge q and velocity in an electromagnetic field is [ Assignment 4 ]:
The spatial part of is the Lorentz force and the time part is the rate of work by this force.
By writing , Maxwell's equations give [ Assignment 4 ]:
This is the energy momentum tensor of the electromagnetic field. Note that is symmetric as required and the energy density is [ Assignment 4 ]