1Preliminaries

IB Electromagnetism

1.1 Charge and Current

The strength of the electromagnetic force experienced by a particle is determined

by its (electric) charge. The SI unit of charge is the Coulomb. In this course, we

assume that the charge can be any real number. However, at the fundamental

level, charge is quantised. All particles carry charge

q

=

ne

for some integer

n

,

and the basic unit

e ≈ 1.6 × 10

−19

C

. For example, the electron has

n

=

−

1,

proton has n = +1, neutron has n = 0.

Often, it will be more useful to talk about charge density ρ(x, t).

Definition

(Charge density)

.

The charge density is the charge per unit volume.

The total charge in a region V is

Q =

Z

V

ρ(x, t) dV

When we study charged sheets or lines, the charge density would be charge

per unit area or length instead, but this would be clear from context.

The motion of charge is described by the current density J(x, t).

Definition (Current and current density). For any surface S, the integral

I =

Z

S

J ·dS

counts the charge per unit time passing through

S

.

I

is the current, and

J

is

the current density, “current per unit area”.

Intuitively, if the charge distribution

ρ

(

x, t

) has velocity

v

(

x, t

), then (ne-

glecting relativistic effects), we have

J = ρv.

Example.

A wire is a cylinder of cross-sectional area

A

. Suppose there are

n

electrons per unit volume. Then

ρ = nq = −ne

J = nqv

I = nqvA.

It is well known that charge is conserved — we cannot create or destroy

charge. However, the conservation of charge does not simply say that “the total

charge in the universe does not change”. We want to rule out scenarios where a

charge on Earth disappears, and instantaneously appears on the Moon. So what

we really want to say is that charge is conserved locally: if it disappears here, it

must have moved to somewhere nearby. Alternatively, charge density can only

change due to continuous currents. This is captured by the continuity equation:

Law (Continuity equation).

∂ρ

∂t

+ ∇· J = 0.

We can write this into a more intuitive integral form via the divergence

theorem.

The charge Q in some region V is defined to be

Q =

Z

V

ρ dV.

So

dQ

dt

=

Z

V

∂ρ

∂t

dV = −

Z

V

∇ ·J dV = −

Z

S

J ·dS.

Hence the continuity equation states that the change in total charge in a volume

is given by the total current passing through its boundary.

In particular, we can take

V

=

R

3

, the whole of space. If there are no

currents at infinity, then

dQ

dt

= 0

So the continuity equation implies the conservation of charge.