Page 29 - Electromagnetics Handbook

Note that the charge density decreases with time less rapidly for a moving observer than
for a stationary one (3/4 as fast):the moving observer is following the charge outward,
and ρ ∝ r. Now we can check the continuity equations. First we see
Dρ 3 −βt −βt
3
+ ρ∇· v =− βρ 0 re + (ρ 0 re ) β = 0,
Dt 4 4
as required for a moving observer; second we see
∂ρ −βt −βt
+∇ · J =−βρ 0 re + βρ 0 e = 0,
∂t
as required for a stationary observer.

The continuity equation in fewer dimensions. The continuity equation can also
be used to relate current and charge on a surface or along a line. By conservation of
charge we can write
d
ρ s (r, t) dS =− J s (r, t) · ˆ m dl (1.14)
dt S
where ˆ m is the vector normal to the curve and tangential to the surface S. By the
surface divergence theorem (B.20), the corresponding point form is

∂ρ s (r, t)
+∇ s · J s (r, t) = 0. (1.15)
∂t
Here ∇ s · J s is the surface divergence of the vector field J s . For instance, in rectangular
coordinates in the z = 0 plane we have

∂ J sx ∂ J sy
∇ s · J s = + .
∂x ∂y
In cylindrical coordinates on the cylinder ρ = a, we would have
1 ∂ J sφ ∂ J sz
∇ s · J s = + .
a ∂φ ∂z
A detailed description of vector operations on a surface may be found in Tai [190], while
many identities may be found in Van Bladel [202].
The equation of continuity for a line is easily established by reference to Figure 1.2.
Here the net charge exiting the surface during time t is given by
t[I (u 2 , t) − I (u 1 , t)].

Thus, the rate of net increase of charge within the system is
dQ(t) d
= ρ l (r, t) dl =−[I (u 2 , t) − I (u 1 , t)]. (1.16)
dt dt
The corresponding point form is found by letting the length of the curve approach zero:
∂ I (l, t) ∂ρ l (l, t)
+ = 0, (1.17)
∂l ∂t
where l is arc length along the curve. As an example, suppose the line current on a
circular loop antenna is approximately
ωa

I (φ, t) = I 0 cos φ cos ωt,
c

© 2001 by CRC Press LLC