In this sense, the terms electric and magnetic “ﬂux densities” for the quantities
D
,
B
are somewhat of a
misnomer because they do not represent anything that ﬂows.
10
1.
Maxwell’s Equations
Fig. 1.6.1
Flux of a quantity.
Similarly, when
J
represents momentum ﬂux, we expect to have
J
mom
=
cρ
mom
.
Momentum ﬂux is defined as
J
mom
=
Δp/(ΔSΔt)
=
ΔF/ΔS
, where
p
denotes momen-
tum and
ΔF
=
Δp/Δt
is the rate of change of momentum, or the force, exerted on the
surface
ΔS
. Thus,
J
mom
represents force per unit area, or pressure.
Electromagnetic waves incident on material surfaces exert pressure (known as ra-
diation pressure), which can be calculated from the momentum ﬂux vector. It can be
shown that the momentum ﬂux is numerically equal to the energy density of a wave, that
is,
J
mom
=
ρ
en
, which implies that
ρ
en
=
ρ
mom
c
. This is consistent with the theory of
relativity, which states that the energy-momentum relationship for a photon is
E
=
pc
.
1.7
Charge Conservation
Maxwell added the displacement current term to Amp`
ere’s law in order to guarantee
charge conservation. Indeed, taking the divergence of both sides of Amp`
ere’s law and
using Gauss’s law
∇ ·
D
=
ρ
, we get:
∇ ·
∇ ×
H
=
∇ ·
J
+
∇ ·
∂
D
∂t
=
∇ ·
J
+
∂
∂t
∇ ·
D
=
∇ ·
J
+
∂ρ
∂t
Using the vector identity
∇·
∇×
H
=
0, we obtain the differential form of the charge
conservation law:
∂ρ
∂t
+
∇ ·
J
=
0
(charge conservation)
(1.7.1)
Integrating both sides over a closed volume
V
surrounded by the surface
S
, as
shown in Fig. 1.7.1, and using the divergence theorem, we obtain the integrated form of
Eq. (1.7.1):
S
J
·
d
S
= −
d
dt
V
ρ dV
(1.7.2)
The left-hand side represents the total amount of charge ﬂowing
outwards
through
the surface
S
per unit time. The right-hand side represents the amount by which the
charge is
decreasing
inside the volume
V
per unit time. In other words, charge does not
disappear into (or created out of) nothingness—it decreases in a region of space only
because it ﬂows into other regions.
Another consequence of Eq. (1.7.1) is that in good conductors, there cannot be any
accumulated
volume
charge.
Any such charge will quickly move to the conductor’s
surface and distribute itself such that to make the surface into an equipotential surface.

1.8.
Energy Flux and Energy Conservation
11
Fig. 1.7.1
Flux outwards through surface.
Assuming that inside the conductor we have
D
=
E
and
J
=
σ
E
, we obtain
∇ ·
J
=
σ
∇ ·
E
=
σ
∇ ·
D
=
σ
ρ
∂ρ
∂t
+
σ
ρ
=
0
(1.7.3)
with solution:
ρ(
r
, t)
=
ρ
0
(
r
)e
−
σt/
where
ρ
0
(
r
)
is the initial volume charge distribution. The solution shows that the vol-
ume charge disappears from inside and therefore it must accumulate on the surface of
the conductor. The “relaxation” time constant
τ
rel
=
/σ
is extremely short for good
conductors. For example, in copper,
τ
rel
=
σ
=
8
.

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- Spring '14
- Energy, Electric charge, Permittivity, Dielectric, unit volume