1
PHYS809Class30Notes
Conservation of energy for electromagnetic fields
We begin by considering electromagnetic waves propagating in vacuum. The effects of linear media will
be considered later. At a fundamental level the waves are due to motion of charged particles. Consider
such a particle of charge
q
. The force of the electric and magnetic fields acting on this particle is the
Lorentz force
(
)
.
q
=
+
×
F
E
v
B
(30.1)
The rate at which this force does work on the particle is
.
q
⋅
=
⋅
v F
v E
(30.2)
By considering all the particles in unit volume, we see that the work done by the fields per unit volume
per unit time is
,
⋅
J E
(30.3)
where
J
is the current density. This work goes into accelerating the changes and converts energy from
the electromagnetic fields into kinetic energy. The rate at which work is done to change the energy of
the fields in a fixed volume
V
is
.
V
dW
dV
dt
= 
⋅
∫
J E
(30.4)
We can use a Maxwell equation to eliminate the current density
0
0
1
.
V
dW
dV
dt
t
ε
μ
∂
= 
∇×

⋅
∂
∫
E
B
E
(30.5)
Using the identity
(
)
,
⋅∇×
= ∇⋅
×
+
⋅∇×
E
B
B
E
B
E
(30.6)
This becomes
(
)
0
0
0
1
1
.
V
dW
dV
dt
t
ε
μ
μ
∂
= 
∇⋅
×
+
⋅∇×

⋅
∂
∫
E
B
E
B
E
E
(30.7)
Using another Maxwell equation to eliminate
∇×
E
, we get
(
)
0
0
0
1
1
.
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 Fall '11
 MacDonald
 Conservation Of Energy, Energy, Magnetic Field, ∂t, Energy density

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