Page 148 - Engineering Electromagnetics, 8th Edition

130 ENGINEERING ELECTROMAGNETICS

This last relationship has some resemblance to Gauss’s law, and we may now gener-
alize our definition of electric flux density so that it applies to media other than free
space. We first write Gauss’s law in terms of 0 E and Q T , the total enclosed charge,
bound plus free:

0 E · dS (23)
Q T =
S
where
Q T = Q b + Q
and Q is the total free charge enclosed by the surface S. Note that the free charge
appears without a subscript because it is the most important type of charge and will
appear in Maxwell’s equations.
Combining these last three equations, we obtain an expression for the free charge
enclosed,

( 0 E + P) · dS (24)
Q = Q T − Q b =
S
D is now defined in more general terms than was done in Chapter 3,

D = 0 E + P (25)
There is thus an added term to D that appears when polarizable material is present.
Thus,

D · dS (26)
Q =
S
where Q is the free charge enclosed.
Utilizing the several volume charge densities, we have

Q b = ρ b dv
ν

Q = ρ ν dv
ν

Q T = ρ T dv
ν
With the help of the divergence theorem, we may therefore transform Eqs. (22), (23),
and (26) into the equivalent divergence relationships,

∇ · P =−ρ b
∇ · 0 E = ρ T
(27)
∇ · D = ρ ν
We will emphasize only Eq. (26) and (27), the two expressions involving the free
charge, in the work that follows.