h/Close to the surface, the
relationship between E and σ
is a fixed one, regardless of
the geometry. The flea can’t
determine the size or shape of
her world by comparingEandσ.the plates are a metal conductor, not an insulator, so the charge
will tend to arrange itself more densely near the edges, rather than
spreading itself uniformly on each plate. Furthermore, we have only
calculated theon-axis field in example 15; in the off-axis region,
each disk’s contribution to the field will be weaker, and it will also
point away from the axis a little. But if we are willing to ignore
these complications for the sake of a rough analysis, then the fields
superimpose as shown in figure f: the fields cancel the outside of
the capacitor, but between the plates its value is double that con-
tributed by a single plate. This cancellation on the outside is a very
useful property for a practical capacitor. For instance, if you look at
the printed circuit board in a typical piece of consumer electronics,
there are many capacitors, often placed fairly close together. If their
exterior fields didn’t cancel out nicely, then each capacitor would in-
teract with its neighbors in a complicated way, and the behavior of
the circuit would depend on the exact physical layout, since the
interaction would be stronger or weaker depending on distance. In
reality, a capacitor does create weak external electric fields, but their
effects are often negligible, and we can then use thelumped-circuit
approximation, which states that each component’s behavior de-
pends only on the currents that flow in and out of it, not on the
interaction of its fields with the other components.10.3.2 The field near a charged surface
From a theoretical point of view, there is something even more
intriguing about example 15: the magnitude of the field for small
values ofz(z b) isE = 2πkσ, which doesn’t depend onbat
all for a fixed value ofσ. If we made a disk with twice the radius,
and covered it with the same number of coulombs per square meter
(resulting in a total charge four times as great), the field close to
the disk would be unchanged! That is, a flea living near the center
of the disk, h, would have no way of determining the size of her flat
“planet” by measuring the local field and charge density. (Only by
leaping off the surface into outer space would she be able to measure
fields that were dependent onb. If she traveled very far, toz b,
she would be in the region where the field is well approximated by
|E|≈kQ/z^2 =kπb^2 σ/z^2 , which she could solve forb.)
What is the reason for this surprisingly simple behavior of the
field? Is it a piece of mathematical trivia, true only in this particular
case? What if the shape was a square rather than a circle? In other
words, the flea gets no information about thesizeof the disk from
measuringE, sinceE = 2πkσ, independent ofb, but what if she
didn’t know theshape, either? If the result for a square had some
other geometrical factor in front instead of 2π, then she could tell
which shape it was by measuringE. The surprising mathematical
fact, however, is that the result for a square, indeed for any shape
whatsoever, isE= 2πσk. It doesn’t even matter whether the sur-
Section 10.3 Fields by superposition 601