Topics in Classical Electrodynamics

and, therefore,
⎛
F µν = η µσ η νρ F σρ
⎜
⎝
0 −E x −E y −E z
E x 0 −H z H y
E y H z 0 −H x
E z −H y H x 0
⎞
⎟
⎠ , (38)
where we have defined the F 0i components to be the electric fields and the
F ij components to the magnetic fields. From the electric and magnetic fields
one can make invariants, i.e. objects that remain unchanged under Lorentz
transformations. In terms of the tensor of th electromagnetic field two such
invariants are
F µν F µν = inv ; (39)
ε µνρσ F µν F ρσ = inv . (40)
Let us inspect the gauge invariance of the electric and magnetic fields ⃗ E and
⃗H, which from the form and their in terms of the electromagnetic field tensor
components can be expressed in terms of the vector potential as
⃗E = − ⃗ ∇ϕ − 1 c
∂ ⃗ A
∂t
and ⃗ H = rot ⃗ A . (41)
One can easily see that in the first case an extra ϕ term cancels with an extra
⃗A term and in the second case we have the gauge transformation contribution
vanishing due to the fact that rot gradχ = 0. We look back at the expression
for the Lorentz force and try to write it in terms of electric and magnetic
fields. Rearranging (33), we get
dp i
dt
=
=
( e
c F i0 U 0 + e ) ds
c F ij U j
dt =
⎛
⎞
⎝ e 1
c Ei √ + e √
1 −
c F ij ⃗v
√ ⎠ c 1 − ⃗v2
⃗v2 c 1 −
c . (42)
2 ⃗v2
c 2 c 2
We can thus rewrite the expression for the Lorentz force as
dp i
dt = eEi + e [
⃗v, H
c
⃗ ]
. (43)
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