Special Theory of Relativity
19
Electromagnetic fields are described by Maxwell’s equations. These
equations in rationalised mks units, outside material media, are:
0
ρ
∇ ⋅ = ε
E
(a)
0
∇ ⋅ =
B
(b)
0
t
∂
∇ × +
=
∂
B
E
(c)




 





 
(1.60)
0
2
1
t
c
∂
∇× −
= µ
∂
E
B
J (d)
where E is the electric field, B is the magnetic field, ρ is the charge density, J is
the current density, ε 0 is the capacitivity of vacuum and µ 0 is the
permeability of vacuum. The motion of a charged particle in the presence of
electromagnetic fields is given by
(
)
=
+
d
q
dt
p
E u×B
(1.61)
where q is the charge of the particle, and the expression on the right hand side
is called the Lorentz force.
It is most convenient to describe the electromagnetic fields in terms of the
electromagnetic potentials A and φ. From Eq. (1.60b), it is seen that B is of the
form
B = ∇
∇ ∇
∇ ∇ × A
(1.62)
Substitution of this relation in Eq. (1.60c) then implies that E can be written
in the form
∂
= −
− ∇ φ
∂t
A
E
(1.63)
The remaining two Maxwell’s equations lead to
2
0
(
)
∂
ρ
∇ φ +
∇ ⋅ = −
∂
ε
t
A
2
2
0
2
2
2
1
1
∂
∂ φ


∇
−
−∇ ∇⋅ +
=−µ


∂


t
c dt
c
A
A
A
J
It may be observed that Eqs. (1.62) and (1.63) do not determined A and φ
uniquely. A transformation
→ +∇ Λ
A A
(1.64)
t
∂
φ → φ − Λ
∂
(1.65)
where Λ
Λ Λ
Λ Λ is a scalar function, does not alter B and E. Therefore some subsidiary
conditions can be imposed on A. This is done by requiring that