How might one predict the arrangement of charges on a
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'=R2/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=kqr/r3+kq'r'/r'3. From this the electric field and induced charge-density σ = -εoE 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.