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
0
ˆ
k sin
2
(k · x ωt).
The average over T =
2π
ω
is thus
hSi =
E
2
0
2
0
ˆ
k.