Edge of Innity The Clash between Edge Eect and Innity Assumption for the Distribution of Charge on a Conducting Plate Quy C. Tran1Nam H. Nguyen2Thach A. Nguyen3and Trung V. Phan4

2025-05-03 0 0 593.84KB 5 页 10玖币
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Edge of Infinity: The Clash between Edge Effect and Infinity Assumption
for the Distribution of Charge on a Conducting Plate
Quy C. Tran,1Nam H. Nguyen,2Thach A. Nguyen,3and Trung V. Phan4, ∗
1High School for Gifted Students, Vietnam National University, Hanoi 100000, Vietnam.
2Duke Univeristy, Durham, NC 27708, USA
3Le Hong Phong High School for the Gifted, Ho Chi Minh 700000, Vietnam
4Department of Molecular, Cellular, and Developmental Biology,
Yale University, New Haven, CT 06520, USA
Abstract
We re-examine a familiar problem given in introductory to physics courses, about determining the induce charge distribution
on an uncharged “infinitely-large” conducting plate when placing parallel to it a uniform charged nonconducting plate of the
same size. We show that, no matter how large the plates are, the edge effect will always be strong enough to influence the
charge distribution deep in the central region, which totally destroyed the infinity assumption (that the surface charge densities
on the two sides are uniform and of opposite magnitudes). For a more detail analysis, we solve the Poisson’s equation for a
similar setting in two-dimensional space and obtain the exact charge distribution, helping us to understand what happen how
charge distributes at the central, the asymptotic and the edge regions.
I. A CURIOUS PUZZLE
One of the authors has been teaching in college for
many years, and during that time there is always the
following homework problem (or similar) for the intro-
ductory course to electromagnetic every year:
“An infinite nonconducting plate with uniform surface
charge density +σ∗
0is placed in parallel to an uncharged
conducting plate. Find the induced charge distribution on
both sides of the conducting plate.” See Fig. 1.
FIG. 1: A physical setting for the problem of interests in
three-dimensional space, with the separation Hbetween
these parallel plates are much smaller than their size R.
The official solution to this problem, either given by
the staffs running the course or handed down through out
the years, is to use the infinity assumption so that the
charge density on the side closer to the nonconducting
plate σ↑and on the side further to the nonconducting
plate σ↓are uniform and of equal magnitude but opposite
signs σ↑=−σ↓(for total charge neutrality). Then, from
the condition that there is no electrical field ~
Einside
the conducting plate, the following relation has to be
satisfied:
~
E=σ∗
0
2+σ↑
2−σ↓
2= 0 ⇒σ↑=−σ↓=−σ∗
0
2.(1)
In other words, it is estimated that the surface densities
on both side of the uncharged conducting plate with have
the same magnitude of half the surface density on the
nonconducting plate.
Let us take a step back and try to understand what
is so puzzling about this solution, even though it might
seems perfectly sounded at first. There is no such thing
as infinite plates – only an “infinitely-large” ones can ex-
ist. Say, the (radial) size of the plates in the problem is
Rand the separated distance between them is H, then
the infinity assumption can be provoked when consider-
ing what happen deep in the central region when RH
and. It is physical to say R→ ∞, but it is unphysical to
say R=∞. In other words, there must be a boundary,
an edge to this infinity. While the edge effect typically
contributes the most to the fringe electrical field far away
from the central region, it can influence the charge distri-
bution very drastically. Consider a circular conducting
plate of radius Rhaving total charged Q, what is the sur-
face charge density σ(r) (on both sides) at radial position
rfrom the center of the plate? This is a famous ques-
tion, one of the few in classical physics which can be an-
swered straight-forwardly by adding extra-dimensions1.
J. J. Thomson has given an elegant geometric argument
for the charge distribution on a plate as the limiting case
of an oblate ellipsoid2:
σ(r) = Q
4πR21−r
R−1/2
,(2)
which is also equal to the projected surface charge distri-
bution on the sphere onto its equator’s plane. For a total
charge that scales with the plate’s area, i.e. Q∝R2, at
the central region the charge distribution σ(0) can be a
1
arXiv:2210.13665v2 [physics.class-ph] 3 Nov 2022
摘要：

EdgeofIn nity:TheClashbetweenEdgeE ectandIn nityAssumptionfortheDistributionofChargeonaConductingPlateQuyC.Tran,1NamH.Nguyen,2ThachA.Nguyen,3andTrungV.Phan4,1HighSchoolforGiftedStudents,VietnamNationalUniversity,Hanoi100000,Vietnam.2DukeUniveristy,Durham,NC27708,USA3LeHongPhongHighSchoolfortheGifte...

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