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Chapter 15
Conservation of Energy and
Momentum
We go directly to the law of conservation of energy, mechanical and electromagnetic, for a
system of particles that interact by electromagnetic forces and are also subject to external
electromagnetic ﬁelds. Then we examine more in detail the magnetic energy.
15.1
Energy density and ﬂow
We assume (for the time being) that only electromagnetic forces are present and that particle
motions are conﬁned to a ﬁnite volume
V
. At speeds small compared to
c,
the particles obey
Newton’s equations
m
i
d
v
i
dt
=
q
i
E
(
x
i
)+
v
i
×
B
(
x
i
)
(15.1)
Multiplying both sides by
v
i
and summing over
i
we obtain
X
i
d
dt
±
1
2
m
i
v
2
i
²
=
X
i
q
i
v
i
·
E
(
x
i
)
(15.2)
or
d
dt
U
kin
=
Z
V
J
(
x
)
·
E
(
x
)
d
3
x
(15.3)
where we have introduced the total kinetic energy
U
kin
=
X
i
±
1
2
m
i
v
2
i
²
(15.4)
and the current density
J
(
x
)=
X
i
q
i
v
i
δ
(
x

x
i
)
(15.5)
Equation (15.3) says that
J
·
E
is the power per unit volume supplied by the ﬁeld to the
particles. We note that
E
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