W9_parallel_resonance.eps

148 Week 4: Capacitance

This form proves to be most useful in the more advanced treatments of electrodynamics that e.g.
physics majors will take that build on this course, but is beyond the scope of this course. It is still
worth reading about in passing for “culture”, or to plant a seed or two that might flower later if you
continue studying physics. If this does not describe you (and it wellmight not!) feel free to skip the
material between the next two separator lines.

We see that the field produced by the usual free charge we considered in the first three chapters
changes form “suddenly” – isdisplaced– at the surface of neutral dielectric materials. It is useful
to define a new field, closely related to the electric field (and force) experienced by a bare test
charge anywhere in space in a medium of some sort or not. We will thinkof this new field as being
producedonlyby bare unbalanced charge, andexplicitly excludefrom consideration the “bound”
neutral charge that we have been discussing above. We will call thisnon-bound chargefreecharge.
This fieldwill not change formas it propagates from one material to another!

The field in question is called theelectric displacement:

D~=ǫE~ (254)

Note well that this is a very odd name. One would be inclined to call thereaction fieldproduced
by the surface bound charge the “displacement” of the vacuum field inside a medium, butthis is
incorrect. On the other hand, the electric displacementdoes not changeat the surface of a
dielectric medium, totally counterintuitively! This drove me batty foryears of study as a physics
major and even as a graduate student because it is some sense an abuse of the English language.

Don’t fight it, accept it! The electric displacement “is what it is” according to this definition, and
is theun-displaced version of the electric field. Sure, it might have been moreuseful and descriptive
to call it the “charge field”, but we are at this point all stuck with thename, so if you plan to go on
in physics you might as well learn it.

The fundamental advantage of this electric displacement (field) is that we can write Gauss’s Law
anywhere, inside a dielectric, conductor, or vacuum, in a form that dependsonlyon the free charge
present, not on any dielectric response of the medium. Since we’ve cancelled out all dependence on
permittivity, this form is just: ∮

S

D~·nˆdA=

∫

V/S

ρfdV (255)

whereρfis the free charge density only. Note theabsenceof any form of the dielectric permittivity!
If we solve this, we can find the resulting field inside any linear medium byjust dividingD~ by
ǫ=ǫrǫ 0.

Following this reasoning, the electric displacement of a point charge iseven simpler than the
electric field of a point charge in charge centered coordinates:

D~=^1

4 π

Q

r^2

rˆ (256)

Note well the absence ofǫ 0! The displacement itself has the units of charge per unit area and
completely captures thegeometryof Gauss’s Law, but it is avectorthat does not correspond in
any way to an actual surface charge density. In some sense it corresponds to theimaginary(as in
pretend, not complex) surface charge density one would get if onetook the central charge,displaced
it uniformlyby a distancer, producing the same charge smeared out uniformly over the spherical
surface of radiusr, and then made it a vector directed outward for positive charge aninwards for
negative charge.

All clear now? Well, probably not so much. Possibly even as clear as mud! But if you think
about it even a bit now, and pay attention to my warnings about the undisplaced displacement field