Page 130 - Engineering Electromagnetics, 8th Edition

112 ENGINEERING ELECTROMAGNETICS

amounts of positive and negative charge may be simultaneously created, obtained by
separation, or lost by recombination.
The continuity equation follows from this principle when we consider any region
bounded by a closed surface. The current through the closed surface is

J · dS
I =
S
and this outward flow of positive charge must be balanced by a decrease of positive
charge (or perhaps an increase of negative charge) within the closed surface. If the
charge inside the closed surface is denoted by Q i , then the rate of decrease is −dQ i /dt
and the principle of conservation of charge requires

dQ i
I = J · dS =− (4)
S dt
It might be well to answer here an often-asked question. “Isn’t there a sign error?
I thought I = dQ/dt.” The presence or absence of a negative sign depends on what
current and charge we consider. In circuit theory we usually associate the current flow
into one terminal of a capacitor with the time rate of increase of charge on that plate.
The current of (4), however, is an outward-flowing current.
Equation (4) is the integral form of the continuity equation; the differential, or
point, form is obtained by using the divergence theorem to change the surface integral
into a volume integral:

J · dS = (∇ · J) dv
S vol
We next represent the enclosed charge Q i by the volume integral of the charge density,
d

(∇ · J) dv =− ρ ν dv
vol dt vol
Ifweagreetokeepthesurfaceconstant,thederivativebecomesapartialderivative
and may appear within the integral,

∂ρ ν
(∇ · J) dv = − dv
vol vol ∂t
from which we have our point form of the continuity equation,
∂ρ ν
(∇ · J) =− (5)
∂t
Remembering the physical interpretation of divergence, this equation indicates
that the current, or charge per second, diverging from a small volume per unit volume
is equal to the time rate of decrease of charge per unit volume at every point.
As a numerical example illustrating some of the concepts from the last two sec-
tions, let us consider a current density that is directed radially outward and decreases
exponentially with time,
1
−t
J = e a r A/m 2
r