5Electromagnetism and relativity

IB Electromagnetism

5.4 Maxwell Equations
To write out Maxwell’s Equations relativistically, we have to write them out in
the language of tensors. It turns out that they are
µ
F
µν
= µ
0
J
ν
µ
˜
F
µν
= 0.
As we said before, if we find the right way of writing equations, they look really
simple! We don’t have to worry ourselves with where the c and µ
0
, ε
0
go!
Note that each law is actually 4 equations, one for each of
ν
= 0
,
1
,
2
,
3.
Under a Lorentz boost, the equations are not invariant individually. Instead,
they all transform nicely by left-multiplication of Λ
ν
ρ
.
We now check that these agree with the Maxwell’s equations.
First work with the first equation: when ν = 0, we are left with
i
F
i0
= µ
0
J
0
,
where i ranges over 1, 2, 3. This is equivalent to saying
·
E
c
= µ
0
ρc,
or
· E = c
2
µ
0
ρ =
ρ
ε
0
When ν = i for some i = 1, 2, 3, we get
µ
F
µi
= µ
0
J
i
.
So after some tedious calculation, we obtain
1
c
t
E
c
+ × B = µ
0
J.
With the second equation, when ν = 0, we have
i
˜
F
i0
= 0 · B = 0.
ν = i gives
µ
˜
F
µi
= 0
B
t
+ × E.
So we recover Maxwell’s equations. Then we now see why the
J
ν
term appears
in the first equation and not the second it tells us that there is only electric
charge, not magnetic charge.
We can derive the continuity equation from Maxwell’s equation here. Since
ν
µ
F
µν
= 0 due to anti-symmetry, we must have
ν
J
ν
= 0. Recall that we
once derived the continuity equation from Maxwell’s equations without using
relativity, which worked but is not as clean as this.
Finally, we recall the long-forgotten potential
A
µ
. If we define
F
µν
in terms
of it:
F
µν
=
µ
A
ν
ν
A
µ
,
then the equation
µ
˜
F
µν
µ
˜
F
µν
=
1
2
ε
µνρσ
µ
F
ρσ
=
1
2
ε
µνρσ
µ
(
ρ
A
σ
σ
A
ρ
) = 0
by symmetry. This means that we can also write the Maxwell equations as
µ
F
µν
= µ
0
J
ν
where F
µν
=
µ
A
ν
ν
A
µ
.