Page 303 - Engineering Electromagnetics, 8th Edition

CHAPTER 9 Time-Varying Fields and Maxwell’s Equations 285

Replacing ρ ν with ∇ · D,
∂ ∂D
∇ · G = (∇ · D) =∇ ·
∂t ∂t
from which we obtain the simplest solution for G,
∂D
G =
∂t
Amp`ere’s circuital law in point form therefore becomes
∂D
∇× H = J + (17)
∂t
Equation (17) has not been derived. It is merely a form we have obtained that
does not disagree with the continuity equation. It is also consistent with all our other
results, and we accept it as we did each experimental law and the equations derived
from it. We are building a theory, and we have every right to our equations until they
are proved wrong. This has not yet been done.
We now have a second one of Maxwell’s equations and shall investigate its sig-
nificance. The additional term ∂D/∂t has the dimensions of current density, amperes
per square meter. Because it results from a time-varying electric flux density (or dis-
placement density), Maxwell termed it a displacement current density.We sometimes
denote it by J d :
∇× H = J + J d
∂D
J d =
∂t
This is the third type of current density we have met. Conduction current density,

J = σE
is the motion of charge (usually electrons) in a region of zero net charge density, and
convection current density,

J = ρ ν v
is the motion of volume charge density. Both are represented by J in (17). Bound
current density is, of course, included in H.Ina nonconducting medium in which no
volume charge density is present, J = 0, and then
∂D
∇× H = (if J = 0) (18)
∂t
Notice the symmetry between (18) and (15):
∂B
∇× E =− (15)
∂t
Again, the analogy between the intensity vectors E and H and the flux density
vectors D and B is apparent. We cannot place too much faith in this analogy, however,
for it fails when we investigate forces on particles. The force on a charge is related to E