4Electrodynamics

IB Electromagnetism

4.2 Magnetostatic energy

Suppose that a current

I

flows along a wire

C

. From magnetostatics, we know

that this gives rise to a magnetic field B, and hence a flux Φ given by

Φ =

Z

S

B · dS,

where S is the surface bounded by C.

Definition (Inductance). The inductance of a curve C, defined as

L =

Φ

I

,

is the amount of flux it generates per unit current passing through

C

. This is a

property only of the curve C.

Inductance is something engineers care a lot about, as they need to create

real electric circuits and make things happen. However, us mathematicians find

these applications completely pointless and don’t actually care about inductance.

The only role it will play is in the proof we perform below.

Example

(The solenoid)

.

Consider a solenoid of length

`

and cross-sectional

area A (with `

√

A so we can ignore end effects). We know that

B = µ

0

IN,

where

N

is the number of turns of wire per unit length and

I

is the current. The

flux through a single turn (pretending it is closed) is

Φ

0

= µ

0

IN A.

So the total flux is

Φ = Φ

0

N` = µ

0

IN

2

V,

where V is the volume, A`. So

L = µ

0

N

2

V.

We can use the idea of inductance to compute the energy stored in magnetic

fields. The idea is to compute the work done in building up a current.

As we build the current, the change in current results in a change in magnetic

field. This produces an induced emf that we need work to oppose. The emf is

given by

E = −

dΦ

dt

= −L

dI

dt

.

This opposes the change in current by Lenz’s law. In time

δt

, a charge

Iδt

flows

around C. The work done is

δW = EIδt = −LI

dI

dt

δt.

So

dW

dt

= −LI

dI

dt

= −

1

2

L

dI

2

dt

.

So the work done to build up a current is

W =

1

2

LI

2

=

1

2

IΦ.

Note that we dropped the minus sign because we switched from talking about

the work done by the emf to the work done to oppose the emf.

This work done is identified with the energy stored in the system. Recall

that the vector potential A is given by B = ∇ × A. So

U =

1

2

I

Z

S

B · dS

=

1

2

I

Z

S

(∇ × A) · dS

=

1

2

I

I

C

A · dr

=

1

2

Z

R

3

J · A dV

Using Maxwell’s equation ∇ × B = µ

0

J, we obtain

=

1

2µ

0

Z

(∇ × B) · A dV

=

1

2µ

0

Z

[∇ · (B × A) + B · (∇ × A)] dV

Assuming that

B × A

vanishes sufficiently fast at infinity, the integral of the

first term vanishes. So we are left with

=

1

2µ

0

Z

B · B dV.

So

Proposition. The energy stored in a magnetic field is

U =

1

2µ

0

Z

B · B dV.

In general, the energy stored in E and B is

U =

Z

ε

0

2

E · E +

1

2µ

0

B · B

dV.

Note that while this is true, it does not follow directly from our results for pure

magnetic and pure electric fields. It is entirely plausible that when both are

present, they interact in weird ways that increases the energy stored. However,

it turns out that this does not happen, and this formula is right.