Tensors and Relativity: Chapter 4

The Electromagnetic tensor

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Maxwell's equations for the electromagnetic field [ in units with tex2html_wrap_inline1210 ] are:

eqnarray866

Defining the anti- symmetric tensor tex2html_wrap_inline1212 with components:

eqnarray886

the electric and magnetic fields  are given by

eqnarray888

If we also define a current four- vector :

equation894

Maxwell's equations  can be written as [ Assignment 4 ]

eqnarray898

where tex2html_wrap_inline1214 . We have now expressed Maxwell's equations in tensor form as required by Special Relativity.

The first of these equations implies charge conservaton 

equation900

eqnarray906

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:

equation908

We find [ Assignment 4 ] that tex2html_wrap_inline1220 is unchanged, while

equation913

where tex2html_wrap_inline1222 and tex2html_wrap_inline1224 is the electric field parallel and perpendicular to tex2html_wrap_inline1226 . Thus tex2html_wrap_inline1228 and tex2html_wrap_inline1230 get mixed.

The four- force  on a particle of charge q and velocity tex2html_wrap_inline1234 in an electromagnetic field is [ Assignment 4 ]:

eqnarray927

The spatial part of tex2html_wrap_inline1236 is the Lorentz force  and the time part is the rate of work by this force.

By writing tex2html_wrap_inline1238 , Maxwell's equations give [ Assignment 4 ]:

equation941

where

equation943

This is the energy momentum tensor of the electromagnetic field. Note that tex2html_wrap_inline1240 is symmetric as required and the energy density  is [ Assignment 4 ]

equation945