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Title: Physics/Electromagnetism - The Electromagnetic Field Tensor A page of notes on advanced methods of electromagnetism
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The Electromagnetic tensor

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 equationsin 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 movingwith speed v in the x direction, one can calculate howthe 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 electricfield 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 energydensity  is [ Assignment 4 ] equation945 Peter Dunsby Sat Jun 15 22:02:24 ADT 1996
 

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