How might one predict the arrangement of charges on a
conducting surface,

if the symmetry arguments that make
Gauss's Law so useful are not available?

On the left, you'll find the challenge: Calculate the
negative (cyan) charge-density σ induced on a grounded sphere
of radius R when a positive (red) point charge q is a distance r from
the sphere's center. One possible method is illustrated on the right,
where you'll find depicted
the *image-charge* q'=-q(R/r) at radius r'=R^{2}/r which
sets the electric potential at each point on that
sphere to zero in the presence of the positive charge. Notice in
the animation that
the two charges always lie on a common radial line to the center
of the sphere, and that the image-charge becomes stronger
(brighter cyan) as the red charge approaches the sphere. To figure
out how charges are distributed on the conductor at left,
the electric field at right
from the positive charge q and its negative image-charge q'
were calculated on the sphere surface using
**E**=kq**r**/r^{3}+kq'**r'**/r'^{3}. From this the
electric field and induced
charge-density σ = -ε_{o}E
on the sphere's surface at left was inferred, given that
its electric potential is also held to zero since it is grounded.
This inference works thanks to the uniqueness of electrostatic
field and potential
solutions for any enclosed volume when, along with the position of
charges within, the electrostatic potential is specified
everywhere on its surface.

Questions: What are the surfaces that
bound the "enclosed volume" referred to in the above application example?
How would you solve this same problem if the sphere at left were neutral
and electrically isolated? (*Note:*
Log color
animations of
grounded and neutral spheres are provided below for comparison.)
For example, would a third charge inside the
transparent sphere model (above right) help?
If so, where would you put it and how much charge would it have?
Does the image-charge method work only
in the electrostatic limit,
where rates of movement of the red charge can be ignored?
How would you take into account the magnetic effects of that movement?
The radiation effects? What would you
do if the sphere were a cube or a tetrahedron? Are there robust
numerical platforms for doing these calculations which would
allow one to deal with arbitrary geometries as well as symmetric
solids? Why might
calculations like this be useful e.g. in studies of global
warming, for designers of electromagnetic shielding and
communications, or
in video game physics engines like that by
Havok?

This page is hosted by UM-StL Physics and Astronomy. Thanks to Eric Mandell for the suggestion to display the image-charge model in parallel with the charge density on the sphere, and Ricardo Flores for mention of that radiation applet. What measurements from these simulations can you make as an experimentalist, for comparison to quantitative model predictions? The person responsible for mistakes is P. Fraundorf.