12Maxwell's equations
IA Vector Calculus
12.1 Laws of electromagnetism
Maxwell’s equations are a set of four equations that describe the behaviours
of electromagnetism. Together with the Lorentz force law, these describe all
we know about (classical) electromagnetism. All other results we know are
simply mathematical consequences of these equations. It is thus important to
understand the mathematical properties of these equations.
To begin with, there are two fields that govern electromagnetism, known
as the electric and magnetic field. These are denoted by
E
(
r, t
) and
B
(
r, t
)
respectively.
To understand electromagnetism, we need to understand how these fields
are formed, and how these fields affect charged particles. The second is rather
straightforward, and is given by the Lorentz force law.
Law (Lorentz force law). A point charge q experiences a force of
F = q(E +
˙
r × B).
The dynamics of the field itself is governed by Maxwell’s equations. To state
the equations, we need to introduce two more concepts.
Definition
(Charge and current density)
. ρ
(
r, t
) is the charge density, defined
as the charge per unit volume.
j
(
r, t
) is the current density, defined as the electric current per unit area of
cross section.
Then Maxwell’s equations say
Law (Maxwell’s equations).
∇ · E =
ρ
ε
0
∇ · B = 0
∇ × E +
∂B
∂t
= 0
∇ × B − µ
0
ε
0
∂E
∂t
= µ
0
j,
where
ε
0
is the electric constant (permittivity of free space) and
µ
0
is the
magnetic constant (permeability of free space), which are constants determined
experimentally.
We can quickly derive some properties we know from these four equations.
The conservation of electric charge comes from taking the divergence of the last
equation.
∇ · (∇ × B)
| {z }
=0
−µ
0
ε
0
∂
∂t
(∇ · E)
| {z }
=ρ/ε
0
= µ
0
∇ · j.
So
∂ρ
∂t
+ ∇ · j = 0.
We can also take the volume integral of the first equation to obtain
Z
V
∇ · E dV =
1
ε
0
Z
V
ρ dV =
Q
ε
0
.
By the divergence theorem, we have
Z
S
E · dS =
Q
ε
0
,
which is Gauss’ law for electric fields
We can integrate the second equation to obtain
Z
S
B · dS = 0.
This roughly states that there are no “magnetic charges”.
The remaining Maxwell’s equations also have integral forms. For example,
Z
C=∂S
E · dr =
Z
S
∇ × E dS = −
d
dt
Z
S
B · dS,
where the first equality is from from Stoke’s theorem. This says that a changing
magnetic field produces a current.