Page 306 - Engineering Electromagnetics, 8th Edition

288 ENGINEERING ELECTROMAGNETICS

part of the following drill problem indicates the reason why this additional current
wasnever discovered experimentally.

D9.3. Find the amplitude of the displacement current density: (a) adjacent to
an automobile antenna where the magnetic field intensity of an FM signal is
8
H x = 0.15 cos[3.12(3 × 10 t − y)] A/m; (b)in the air space at a point within a
−6 8
largepower distribution transformer where B = 0.8 cos[1.257×10 (3×10 t−
x)]a y T; (c) within a large, oil-filled power capacitor where r = 5 and E =
√
8
−6
0.9 cos[1.257 × 10 (3 × 10 t − z 5)]a x MV/m; (d)ina metallic conductor
7
at 60 Hz, if = 0 , µ = µ 0 , σ = 5.8 × 10 S/m, and J = sin(377t − 117.1z)a x
2
MA/m .
2
2
2
Ans. 0.468 A/m ; 0.800 A/m ; 0.0150 A/m ; 57.6 pA/m 2
9.3 MAXWELL’S EQUATIONS
IN POINT FORM
We have already obtained two of Maxwell’s equations for time-varying fields,

∂B
∇× E =− (20)
∂t
and
∂D
∇× H = J + (21)
∂t
The remaining two equations are unchanged from their non-time-varying form:

(22)
∇ · D = ρ ν

∇ · B = 0 (23)

Equation (22) essentially states that charge density is a source (or sink) of electric
flux lines. Note that we can no longer say that all electric flux begins and terminates
on charge, because the point form of Faraday’s law (20) shows that E, and hence D,
may have circulation if a changing magnetic field is present. Thus the lines of electric
flux may form closed loops. However, the converse is still true, and every coulomb
of charge must have one coulomb of electric flux diverging from it.
Equation (23) again acknowledges the fact that “magnetic charges,” or poles, are
not known to exist. Magnetic flux is always found in closed loops and never diverges
from a point source.
These four equations form the basis of all electromagnetic theory. They are partial
differential equations and relate the electric and magnetic fields to each other and to