Page 305 - Engineering Electromagnetics, 8th Edition

CHAPTER 9 Time-Varying Fields and Maxwell’s Equations 287

the loop, a magnetic field varying sinusoidally with time is applied to produce an
emf about the closed path (the filament plus the dashed portion between the capacitor
plates), which we shall take as
emf = V 0 cos ωt
Using elementary circuit theory and assuming that the loop has negligible resis-
tance and inductance, we may obtain the current in the loop as
I =−ωCV 0 sin ωt
S
=−ω V 0 sin ωt
d
wherethequantities , S,andd pertaintothecapacitor.LetusapplyAmp`ere’scircuital
law about the smaller closed circular path k and neglect displacement current for the
moment:

H · dL = I k
k
The path and the value of H along the path are both definite quantities (although
difficult to determine), and k H · dL is a definite quantity. The current I k is that
current through every surface whose perimeter is the path k.Ifwe choose a sim-
ple surface punctured by the filament, such as the plane circular surface defined by
the circular path k, the current is evidently the conduction current. Suppose now
we consider the closed path k as the mouth of a paper bag whose bottom passes
between the capacitor plates. The bag is not pierced by the filament, and the con-
ductor current is zero. Now we need to consider displacement current, for within the
capacitor

V 0
D = E = cos ωt
d
and therefore
∂D S
I d = S =−ω V 0 sin ωt
∂t d
This is the same value as that of the conduction current in the filamentary loop.
Therefore the application of Amp`ere’s circuital law, including displacement current
to the path k, leads to a definite value for the line integral of H. This value must be
equal to the total current crossing the chosen surface. For some surfaces the current
is almost entirely conduction current, but for those surfaces passing between the
capacitor plates, the conduction current is zero, and it is the displacement current
which is now equal to the closed line integral of H.
Physically, we should note that a capacitor stores charge and that the electric field
between the capacitor plates is much greater than the small leakage fields outside.
We therefore introduce little error when we neglect displacement current on all those
surfaces which do not pass between the plates.
Displacement current is associated with time-varying electric fields and therefore
exists in all imperfect conductors carrying a time-varying conduction current. The last