Uniform magnetic field in a spherical region

[fysikgubbe]

Homework Statement

Is it possible to design a current winding that will produce a uniform magnetic field in a spherical region of space?

Homework Equations

Well, the literature tells that the equation for the B-field for a uniformly magnetized sphere = (2/3)u_0 * M and also the magnetization current density = J = curlM may be helpful.

The Attempt at a Solution

The truth is that I don't even have a clue where to begin or how to think. Really need some help on this one.

 

[ideasrule]

What do you mean by "a uniform magnetic field in a spherical region of space"? Does the field have to be 0 outside the spherical region or something? If you just want a uniform field, a solenoid or Helmholtz coil can easily provide one. Define an arbitrary but small spherical region and you'd have a uniform field in a spherical region of space.

 

[marcusl]

Neither a finite-length solenoid nor a Helmholtz coil produce a perfectly uniform field, even though both are widely used because their fields are uniform enough for many practical applications.

It is possible in theory to generate a perfectly uniform B field from currents on the surface of a sphere. I think you made a good start by realizing that a magnetized sphere has uniform B inside. The current density equation you wrote down, though, is appropriate for current inside the volume and you will need to deal with the discontinuity at the surface. This has been done for you already in the form of the boundary condition that gives equivalent surface current density at the material/air boundary. Take a look at that equation.