Advanced Electromagnetism Lecture 13 of 15

March 2, 2012 by M. Fabbrichesi

M. Fabbrichesi , SISSA

Covariance of the Maxwell's equations

In this lessons the definition of four-vectors and of scalar product between four-vectors is given and this leads to the Minkowski four-dimensional space whose metric is ds2abdxadxb=-c2dt2+dx2+dy2+dz2. In particular, it is given the definition of the 4-velocity ua and of the four-momentum pa. Next, a few homeworks are suggested to students, e.g. demonstrate that E2=c2(m2c2+p2). Next the acceleration four-vector is introduced by discussing an example: a rocket with g acceleration. In particular, the solution of the uniformly accelerated motion is found.

Next, the Lorentz's force is written in a covariant form by introducing a 4-tensor Fαβ, i.e. the electromagnetic 4-tensor. The components of this antisymmetric tensor are given by the components of the electric and magnetic fields. This allows us to write Maxwell's equations in a very compact form [Jackson sec. 11.9]. Moreover this result together with the definitions of the 4-vectors Jα and Aα establishes the covariance of the equations of electromagnetism.

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