Re: Charge Distribution around pointed areas

[Allen Miller <amiller@SUHEP.PHY.SYR.EDU>]

Here is a non-rigorous plausibility argument as to why charge on a
conductor tends to become concentrated at a sharp point:

The student is expected to know or at least believe three facts:
(1) Electric field lines cannot start or stop in empty space.
(2) Just outside a conductor, the electric field is perpendicular to the
surface of a conductor.
(3) The strength of the electric field just outside a conductor is
proportional to the surface charge density at the corresponding point on
the conductor.

Draw two large parallel plates that have opposite charge on them, but of
equal magnitude. This is the standard parallel-plate capacitor problem.
The surface-charge density on both will be uniform. The electric field in
the region between the plates is uniform. The electric field lines start
from the positively charged plate and terminate at the negatively charged
plate, and are perpendicular to the plates. (All the preceding statements
ignore fringe effects, which are of concern only near the edges.)

Now, put a pointed projection (also a conductor) on, say, the negatively
charged plate. A simple way is to draw a triangle whose base is on the
plate, which projects into some of the space between the plates.

The electric field lines must be normal to the surface of this
projection, in the region close to it. There is no way that they can do
this, unless they curve toward the projection, as they go from the
positive plate to the negative plate.

The field line that intersects the apex of the triangle remains straight.
The field lines on both sides of this field line must curve towards the
projection to maintain perpendicularity.

Hence, there is a concentration (increased density) of field lines in the
vicinity of the projection. The conclusion is that the electric field is
stronger just outside the projection, than it is for points just outside
the flat part of the negatively charged plate.

It follows that the surface charge density is larger on the projection
than it is on the rest of the plate.

Allen Miller, Syracuse University
(315) 443-5962








 On Sun, 14 Nov 1999, Brian Rintoul wrote:

> I'm about to start my first time teaching the unit on Electricity, for the
> Ontario grade 12A Physics course, which opens with electrostatics.
>
> If a charged object is spherical, the charge is uniformly distributed over
> the surface, the reason for this distribution is quite clear.  What isn't
> immediately apparent, this Biochemist is, why charge becomes concentrated
> around sharp pointed areas on a conductor's surface.
>
> Explanation please.
>
>                 \\\|///
>                 [ @ @ ]
> ............o00o//(_)\\o00o...............
> Brian Rintoul
> Evening School Teacher (Ottawa Carleton District School Board, Tuesdays
> and Thursdays)/Occasional Teacher (Ottawa Carleton District School Board,
> Ottawa Carleton Catholic School Board - M.F. McHugh School,
> Fridays)/Ottawa Torah Institute (Mondays - Thursdays, Semester I)
> e-mail: brian_rintoul@ocdsb.edu.on.ca
>
> I thank whoever it was on this Listserv, or a chemistry Listserv I
> subscribe to, that made the graphic I have only slightly altered.  It has
> inspired both positive comments, and variations.