4Electrodynamics

IB Electromagnetism

4.6 Poynting vector

Electromagnetic waves carry energy — that’s how the Sun heats up the Earth!

We will compute how much.

The energy stored in a field in a volume V is

U =

Z

V

ε

0

2

E · E +

1

2µ

0

B · B

dV.

We have

dU

dt

=

Z

V

ε

0

E ·

∂E

∂t

+

1

µ

0

B ·

∂B

∂t

dV

=

Z

V

1

µ

0

E · (∇ × B) − E · J −

1

µ

0

B · (∇ × E)

dV.

But

E · (∇ × B) − B · (∇ × E) = ∇ · (E × B),

by vector identities. So

dU

dt

= −

Z

V

J · E dV −

1

µ

0

Z

S

(E × B) · dS.

Recall that the work done on a particle

q

moving with velocity

v

is

δW

=

qv·E δt

.

So the

J · E

term is the work done on a charged particles in

V

. We can thus

write

Theorem (Poynting theorem).

dU

dt

+

Z

V

J · E dV

| {z }

Total change of energy in V (fields + particles)

= −

1

µ

0

Z

S

(E × B) · dS

| {z }

Energy that escapes through the surface S

.

Definition (Poynting vector). The Poynting vector is

S =

1

µ

0

E × B.

The Poynting vector characterizes the energy transfer.

For a linearly polarized wave,

E = E

0

sin(k · x − ωt)

B =

1

c

(

ˆ

k × E

0

) sin(k · x − ωt).

So

S =

E

2

0

cµ

0

ˆ

k sin

2

(k · x − ωt).

The average over T =

2π

ω

is thus

hSi =

E

2

0

2cµ

0

ˆ

k.