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2 Basic Electromagnetism
charge q, density n, and velocity v, the current density is given by i = nqv. Hence,
using the Coulomb force F = qE, the term i · E in the third term is rewritten as
i · E = n F · v.
(2.55)
Thus, the third term of (2.54) is the work done by the Coulomb force. The variation
in the electromagnetic energy in space V with time is given by
∂U
∂t
=
∂
∂t
V
udV =
V
E ·
∂E
∂t
+
B
μ
·
∂B
∂t
+ i · E
dV .
(2.56)
Using (2.42) and (2.53), the sum of the first and third terms in (2.56) leads to
E ·
∂E
∂t
+ i · E = E ·
∂D
∂t
+ i
= E · (∇ × H).
(2.57)
Using (2.45) and (2.49), the second term becomes
B
μ
·
∂B
∂t
= −H · (∇ × E).
(2.58)
Thus, the right side of (2.56) is rewritten as
V
[E · (∇ × H) − H · (∇ × E)]dV = −
V
∇ · (E × H)dV .
(2.59)
Here, Poynting’s vector is defined as
S P = E × H.
(2.60)
Then, using Gauss’ theorem (2.56) becomes
∂
∂t
V
udV +
S
S P · dS = 0,
(2.61)
where V is the surface of region S. This equation describes the flow of energy: the
variation in energy with time is equal to the sum of Poynting’s vector penetrating
through the surface.
2.5 Maxwell’s Equations and Breaking of Symmetry
Maxwell’s equations that describe electromagnetic phenomena are: