Phys Educ. 24 119891. Printed In the UK
Why does charge concentrate on
points?
H S Fricker
A pointed conductor is modelled by a sequence
of touching spheres. The requirement that the
electric fieldstrength should vanish between
them yields an explanation of charge concentration that appears more realistic than the usual
textbook argument.
Why does charge concentrate on points? When
I am
faced with this question as a teacher
I usually find
myself resorting to rather vague statements about
charges wanting to get as far apart as possible.
so
accumulating at the place
which is most remote from
therest
of theconductorsurface.Thiskind
of
answer is all right as far as it goes and, delivered
with plenty of hand waving, seems to satisfy most
students. However, we ought to be able to do better
if the occasion demands. Exact solutions do
exist for
certain specially shaped conductors, such as spheroids(MorseandFeshbach
1953). However,the
method of solution-separating
variables in the
Laplace equation-tends to obscure the underlying
physics, and the charge concentration appears as if
by magic. What we really want is a simple model,
basedontheinverse-squarelaw,
which shows
clearly why the effect must happen and why it is so
sensitive to the sharpnessof the point. Without such
a model, we cannot really claim to understand what
is going on.
Theonlysimpletreatment
I haveseen in textbooks is onebasedontheproperties
of isolated
I
Figure 1 Charge distribution on apear-shaped
conductor
spheres (see for example Feynman
et a1 1964, Bolton
1974). Although it does predict charge concentration, I believethatthisapproach
is physically
unsound. The object of this article is to explain why
and then describe a different modelwhich I think is
more convincing.
Isolated sphere models
Simple electrostatics shows that,in order to raise an
isolated conducting sphere of radius r to a potential
V , it must have a charge density U , where
U=
Vir.
So for a given V
Hugh Fricker obtained his BA and PhD from the
University of Cambridge and was IC1 Research
Fellow at the University of Manchester from 1970 to
1972. He is a physics teacher at Bradford Grammar
School and is also responsible for higher education
advice and applications. His interests include solid
state theory, quantum chemistry and polymer
theory.
0031-9120~89/030157CO5 $02 5 0 0 1989 IOP Publlshlng Ltd
One approach to the problem of charge concentration(Feynman et a1 1964) is tomodelthe classic
pear-shapedconductor
(figure 1) by a pair of
spheres joined by a thin wire (figure 2 ) . The small
157
If we consider asmall element of the perimeter,of
length dl, a distance d from P, the area generatedby
rotating it about the axis PQ is 2nd sin 8 dl; so if its
charge density is a, its contribution to the potential
at P is
bV= a 2 n d sin OdN4nE,rd
= a sin 8 dN2~,,.
Therefore the total potential at P is
Pear-shaped conductor represented by two
linked spheres
Figure 2
sphere represents the point, the large one the rest
of
the conductor, and the wireallows charge to pass
between them until the potentials equalise.
Zf the
spheres can be considered isolated. so that neither
affects the potential of the other, then equation (1)
shows that their charge densities are inversely proportional to their radii. Since the point has a small
radius, its charge density will be large.
Thisargument is simpleandmemorable.The
trouble is that the spheres can only be considered
truly isolated if they are far apart compared
with
boththeirradii,andapear-shapedconductor
is
nothing like that: its point is close to the bulk of the
conductor and joined to it, not by a negligible wire,
but by asubstantialneck
of metal. To apply the
model we must treat these differences as unimportant. This amounts to assuming that the potential at
apoint
is largely a local matter,dominated by
contributionsfromnearbychargesand
relatively
insensitive to those further away.
However, it is easy to show that potential does not
behave like this. Consider. for instance, the conductor shown in figure 3. For simplicity, it is assumed to
have cylindrical symmetry about theaxis PQ. but its
shape is otherwise arbitrary. What is the potential at
the point P?
Figure 3 Geometry for calculating the potential at
the tip of a pear-shaped conductor
where the line integral is taken along the perimeter
from P to Q.
It is apparent from equation ( 2 ) that the potential
is not in any way dominated by contributions from
thesurfacenearto
P. Thelatterreceiveagentle
weighting from the sin 8 factor, but all parts of the
surface make asignificant contribution. The isolated
sphere
model,
which ignores
these
long-range
effects, is therefore highly suspect.
What
equation
(2) does
demonstrate
is the
importance of the overallsize of theconductor.
Since the integral has dimensions 'charge density X
length' a small conductor, whatever its shape, needs
a large density to produce any given potential. This
is the real significance of equation (1) for a sphere.
The high densityona small sphere is due. not to
strongcurvature in theimmediate vicinity of the
point of interest, but to the smallness of the entire
surface. Once this is realised. the irrelevance of (1)
to a pear-shaped conductor becomes apparent.
An improved model
To progress we must abandon isolated spheres and
model. at least crudely, an actual tapering conductor. Consider a series of N touching spheres of radii
r , , k r , , k 2 r l . . . . k"-lrl (figure 4). By making the
sphere radii decrease by a constant factor k ( < l ) ,
we can simulate a conical point whose semi-vertical
angle p is easily shown to be
~
p=sin"[(l-k)l(l+k)].
If k << 1, p is large and we have a sharply tapering
point, whilst avalue of k close to 1 gives along
slender point with p small. The 'blunt' and 'sharp'
ends of our conductor have radii of r , and k"'r,
respectively.
Tomakethecalculationsmanageable
we shall
assumethateachsphere
is uniformly charged, so
that its external field is that of a point charge at the
centre: but the sphere charges, Q,. Q?, . . . , Q % will
.
be adjusted so that the total electric field strength is
158
zero at each of the points like PI, P*, . . . between
adjacent spheres. This models crudely the process
by which charge on a real conductor distributes
itself
until the internal electric field is zero.
Consider now the pointP, between spheresn and
( n + 1) (figure 5 ) . Theconditionforzero
field
strength at P, is
('
4X&n
...
+
+ (r,-2+2r,-1+2r,)?
(rn-,+2r,)?
r;
Qn - 2
determine the ratios of the U ; their absolute values
will depend, of course, on thepotential of the
conductor.
For general values of k , the equations are hard to
solve.However,anapproximatesolutioncanbe
found for the case of a sharply tapering point, for
which p is large and k small. The simplest, although
very crude, approximationis to put k = 0 in equation
(3), giving
Ul+U?+
=-(-+1
Q,+I
Qn+?
474, r ; , ,
(2rn+,
+r,+$
Q, + 3
+ . . .).
+ 2r,+l+ r,,-#
+
The series on the left- and right-hand sides extend
backtothe
first sphereandforwardtotheNth
sphere respectively.
By representingthechargedensity
QJ4nr; on
sphere n by unr and
recalling
that
the
radius
decreases by the factor k from one sphere to the
next, we can rewrite this in the form
...+
On-?
, ,+-
(1+2k+2k-)-
on-l
(1+2k)'i"
.,.
(4)
+U,=a"+l.
This has a simple physical interpretation. Since. for
small k , the spheres decrease rapidly in radius, we
can regard the pointP, as being 'close to the surface'
of every sphere to the left,
so that each contributes a
field strength of simply U/&,,, whereas the spheres on
the right beyond sphere ( n + 1) are assumed to be
too small to make asignificant contribution. The last
point is not self-evident, because we expect the U to
increase rapidly with n . which could compensate for
the decreasing surface area. However, we shall assume it for the moment and check for consistency
later.
The set of equations like (3) now becomes simply
U? = U1
+ U?
U, = U , + U? + U?
(5)
U3 = U,
k'o, + z
= U n + I +( 2 + k)'
+
k4un+ 3
+ . . . . (3)
( 2 2k k')?
+ +
There are (N- 1) such equations, one for each of
the (N- 1) points between spheres. Together they
etc, whose solution for n 2 2 is easily shown to be
U"
= 2"-%).
(6)
Figure 4 Pointed
conductor represented
by a sequence of
touching spheres
etc
Figure 5 Geometryfor
calculating the electric
field strength a t the
point P,,
159
Wecannowchecktheconsistency
of ignoring
contributions from spheres( n 2) onwards in equation (3). Using equation (6) the right-hand side of
equation (3) becomes
dramatically, on the slender points
usually met in
practice.
Thecalculations in the last sectionthrowsome
further light on the ‘isolated sphere’ treatment discussedearlier.Oneconsequence
of takingthat
2k’
2’k‘
model seriously is a belief that strong surface curvaUntl l+-( 2 + k ) ’ + ( 2 + 2 k + k 2 ) ’ + ’ ”
ture is in itself sufficient to cause dramatic charge
concentration.
It
is sometimes
suggested,
for
and if k is small, the terms after the first are clearly
instance,thata tiny hemisphericalpimpleonthe
negligible.
dome of a Van de Graaff generator would reduce
by
Equation (6) shows
how
the
charge
density
alargefactorthemaximumpotentialto
which it
increases as we move out towards the point (increas- could be raised. This situation can be simulated,
in
ing n ) . For the sphere at the tip
our model, by assumingthattherearejusttwo
touching spheres ( N = 2 ) , o n e large and the other
U,, =2“?u,.
very small. But according to equation
( 5 ) , 0, = U ? ,
implying no chargeconcentrationonthe
small
To relate this to the tip radius, we recall that
sphere, however small its radius. This precise result
rl.=k~v-lr,.
is, of course, an artefact of the approximations we
havemade,butthegeneralconclusion
is not:a
Eliminating N between (7) and (8) gives
region of small radius of curvature will only concenln(2u,~Jul) In 2
trate charge strongly if it is the tip of a proper point
-and projects a distancewhich is large compared with
In(rv/rl) Ink
its radius. As an independent checkof this, consider
aconductinghemispherestuckontoan
infinite
01
charged conducting plane. This case can be solved
U,%~=
/ U+~( r N / r l ) - y
exactly by separating variables in the Laplace equation: it is simply ‘half’thefamiliarproblem
of a
where
conducting sphere placed in a uniform electric field
(seeforexampleBleaneyandBleaney
y = - In21111 k .
1965). We
find that the charge density at the outermost point
of
(Note that since k < 1, In k < 0, making y positive.)
the hemisphere is exactly threetimes greater than
Thus
thatontheremoteparts
of theplane.There
is
therefore
some
charge
concentration.
but
it
is
entirutlpx rt;<.
(11)
ely independent of the hemisphere radius.
The charge density at the tip increases as an inverse
Returningtothegeneralcase
of N touching
power y of the tip radius, where y depends on the
spheres. it is interesting to see how the charge on
angle of taper of the point.
each contributes to the
potential at the right-hand
As an example of the predictions of the model,
tip. A sphere of radius r and charge density U has a
considerasharplypointedconductorfor
which
surface potential of ad&,,.If we again assume small
r,\/rI = 10” and k = 0.1 ( p = 55”). Equation (10)
k , the right-hand tip can be considered ‘close’ to the
gives y = 0.30 and equation (9) predicts that u,%/ul
=
surface of everyone of thespheres, so that its
16. So eventhoughthisconductor
is arapidly
potential is approximately
tapering one, whose point does not project far, the
charge density at the tipis enhanced by quite a large
V t , , - & ~ l ( u , r , + u , r ~. +. . + u , r , ) .
factor.
Usingequation
(6), andrememberingthat
r, =
k”Ir,. we find
Discussion
+
(
In view of the approximations involved it would be
quiteunrealistictoexpecttheresults
of thelast
section to be at all accurate. I feel that the value of
the model lies in the simple physical picture underlying equation (4). Although it appliesonlyto
rapidlytaperingpoints,thesearetheoneswhere
charge concentration is intuitively least likely. Once
we understand whyit happens, it is nothardto
believe that the same thing must occur, and more
160
V,,,= e,
‘glrl(1
+ k + 2k’ + 2?k3+ . . . + 2,”’k””).
Since k is small, the dominant contribution comes
from
largest
the
sphere-the
bulk
the
of
conductor-whilst that from the tip charge (the last
term) is entirely negligible. The tip charge, which is
so vital in cancelling the potential gradient, contributes virtually nothing to the overall potential. This
conclusion is of coursethecompleteopposite
of
what is assumed in the ‘isolated sphere’ model and,
whilstit wouldberelaxedsomewhatforamore
slender point, it does show just how misleading that
model can be.
We conclude by attempting a qualitative explanaton of charge concentration based on the alternative
model presented here. Consider a conductor tapering rapidly to a sharp point. The charge on it must
distribute itself so that the internal electric field is
zero. Therefore at any position P inside the point,
the ‘outward’ field due to the charge behind it must
balance the ‘inward’ field due to the charge further
out. Now imagine P moving outwards towards the
tip. Because of the tapering shape of the conductor,
the ‘outward’ field must increase continuously. The
reason is that, as P moves, it puts more and more
charge ‘close behind’ it, without getting significantly
further away from the bulk of the conductor. This
increase in the ‘outward’field is a purely geometrical
effect, whch wouldoccureven
with auniform
I SCIENCE 89
l
British Association
annual meeting
Sheffield will be the venue for the 151st annual
meeting of the British Association for the
Advancement of Science, taking place on 11-15
September 1989. Among this year’s topics are
‘Measuring the universe’, ‘Quarks’, ‘Genetic
fingerprinting’, ‘Managing science’, ‘Black holes’,
‘New materials’ and ‘Biotechnology’, and young
people will be able to win a prize in the Science
Quest, join in the ‘hands on’ exploratory
exhibition, tax their brains at the problem-solving
workshops, as well as make a video of the
meeting.
There will also be an Interactive Science
Laboratory and exhibitions with industrial
participants, in addition to specialist exhibits in the
University and Polytechnic departments. Seminars
for industry will feature the science and industry
partnership. School parties wishing to attend
‘Science 89’ on a daily basis are particularly
welcome. Full programmes will be available in
July, and details of the meeting are obtainable
now from Dr Connie Martin, Public Affairs
Manager, British Association, Fortress House, 23
Savile Row, London W1X 1AB (tel: 01-494 3326).
chargedensity.However,
it demandsthatthe
balancing ’inward’ field must also increase, and this
canonlyhappen
if thechargedensitybeyondP
increases as P moves outwards. This fact,
in turn,
enhances the growth of the ‘outward’ field, and the
self-consistent charge densitywhich results increases
rapidlyasweapproachthetip
of the point. The
sharper the point, the longer this process continues
and the greater the ultimate charge density.
References
Bleaney B I and Bleaney B 1965 Electricity and
Magnetism (Oxford: Clarendon) section 2.4
Bolton W 1974 Patterns in Physics (London:
McGraw-Hill) p 308
Feynman R P, Leighton R B and Sands M 1964 The
Feynman Lectures on Physics v01 I1 (Reading, MA:
Addison-Wesley) ch 6, p 13
Morse P M and Feshbach H 1953 Methods of Theoretical
Physics (New York: McGraw-Hill) pp 1284-5
I I NEW LIGHT ON TREASURES
I
1
I
A very special exhibition of ancient artefacts will
be on show in York from 1 May until 31 October
1989: entitled ‘Russian holograms - treasure
trapped in light’, the collection represents a wealth
of
rare and beautiful objects-from the museums of
Kiev-which have been translated into holograms.
The exhibits, including jewellery, bowls,
ornaments, head-dresses and regalia in gold, silver,
enamel and ivory, have been selected by the York
Archaeological Trust and cover30 000 years of
Ukrainian history and culture. The holograms
were subsequently produced at the Institute of
Physics of the Ukrainian Academy of Sciences in
Kiev.
Lasers will be used to illuminate the holograms
in a darkened exhibition area at the new
Archaeological Resource Centre in St Saviourgate.
Models, replicas and videos also feature,
combining to produce a fascinating tour of art,
history and archaeology through the medium of
holography. The show will be open seven days a
week from 09.00h and admission is f 2 for adults,
f l for children, with a special rate for parties
which have also booked to visit the Jorvic Viking
Centre.
Further details are available from Linda James,
Information Officer of the York Archaeological
Trust on (0904) 646411, and bookings should be
sent to ‘Russian Holograms’/CRM Ltd, United
House, Piccadilly, York YO1 1PQ.
161