Page 27 - Electromagnetics

potential. This in turn is related to the invariance of Maxwell’s equations under gauge
transforms; the values of the electric and magnetic fields do not depend on which gauge
transformation we use to relate the scalar potential to the vector potential A.
We may state the conservation of charge as follows:
The net charge in any closed system remains constant with time.

This does not mean that individual charges cannot be created or destroyed, only that
the total charge in any isolated system must remain constant. Thus it is possible for a
positron with charge e to annihilate an electron with charge −e without changing the
net charge of the system. Only if a system is not closed can its net charge be altered;
since moving charge constitutes current, we can say that the total charge within a system
depends on the current passing through the surface enclosing the system. This is the
essence of the continuity equation. To derive this important result we consider a closed
system within which the charge remains constant, and apply the Reynolds transport
theorem (see § A.2).

The continuity equation. Consider a region of space occupied by a distribution of
charge whose velocity is given by the vector field v. We surround a portion of charge
by a surface S and let S deform as necessary to “follow” the charge as it moves. Since
S always contains precisely the same charged particles, we have an isolated system for
which the time rate of change of total charge must vanish. An expression for the time
rate of change is given by the Reynolds transport theorem (A.66); we have 2
DQ D ∂ρ
= ρ dV = dV + ρv · dS = 0.
Dt Dt V (t) V (t) ∂t S(t)
The “D/Dt” notation indicates that the volume region V (t) moves with its enclosed
particles. Since ρv represents current density, we can write

∂ρ(r, t)

dV + J(r, t) · dS = 0. (1.10)
∂t
V (t) S(t)
In this large-scale form of the continuity equation, the partial derivative term describes
the time rate of change of the charge density for a fixed spatial position r. At any time t,
the time rate of change of charge density integrated over a volume is exactly compensated
by the total current exiting through the surrounding surface.
We can obtain the continuity equation in point form by applying the divergence the-
orem to the second term of (1.10) to get

∂ρ(r, t)

+∇ · J(r, t) dV = 0.
∂t
V (t)
Since V (t) is arbitrary we can set the integrand to zero to obtain
∂ρ(r, t)
+∇ · J(r, t) = 0. (1.11)
∂t

2 Note that in Appendix A we use the symbol u to represent the velocity of a material and v to represent
the velocity of an artificial surface.

© 2001 by CRC Press LLC