Analyzing Magnetic Field in a Charged Sphere

[Esran]

Imagine a sphere such that positive charge is evenly distributed throughout one hemisphere (not just on the surface) and equal negative charge is evenly distributed throughout the other hemisphere.

Is there a simple or elegant way to map out the magnetic field inside/outside the sphere and predict the behavior of the field (magnitude, direction) at any arbitrary point inside/outside the sphere?

P.S. Ignore the little plus sign outside the sphere, or better yet, pretend it's a test charge. Also, if you can't make heads or tails of how the sphere would work out, then let me know if you have any ideas how a spherical shell with analogous properties would behave.

 

[berkeman]

(you mean electric field, right? Not magnetic field)

 

[Esran]

yeah...that's what lack of sleep does to you.

 

[Pythagorean]

I think you could arrive at a simple elegant diagram by doing all the messy work of figuring the electric field between every charge pair and canceling opposing fields to get rid of arrows and clean up the diagram.

 

[GRDixon]

Imagine a sphere such that positive charge is evenly distributed throughout one hemisphere (not just on the surface) and equal negative charge is evenly distributed throughout the other hemisphere.

Is there a simple or elegant way to map out the magnetic field inside/outside the sphere and predict the behavior of the field (magnitude, direction) at any arbitrary point inside/outside the sphere?

P.S. Ignore the little plus sign outside the sphere, or better yet, pretend it's a test charge. Also, if you can't make heads or tails of how the sphere would work out, then let me know if you have any ideas how a spherical shell with analogous properties would behave.

For points external to the sphere and adequately far away (>>R) the (electric) field would be that of an electric dipole. For other distributions, "The Feynman Lectures on Physics", V2, Sect. 6-5 "The dipole approximation for an arbitrary distribution" is suggested reading. For external points up close to the spherical surface, and/or for points internal to the sphere, I don't know. I'd be inclined to approximate E using a computer.