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Tensors and Relativity: Chapter 4

# The Electromagnetic tensor

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