Electric field inside a dielectric sphere with cavity

[Leotron]

Original Problem:

"A sphere of radius a is made of a nonconducting material that has a uniform volume charge density [PLAIN]http://jkwiens.com/2007/10/24/answer-electric-field-of-a-nonconducting-sphere-with-a-spherical-cavity/d2606be4e0cd2c9a6179c8f2e3547a85_2.gif. A spherical cavity of radius b is removed from sphere which is a distance z from the center of the sphere. Assume that a > z + b. What is the electric field in the cavity?"

I understand how to solve an ordinary problem like this, but what if the sphere has dielectric constant ε which is different from the cavity whose permittivity is ε0? When solving this, should I treat the imaginary small sphere with -ρ to have permittivity ε or ε0?

 

[mfb]

The imaginary sphere has the same material as the sphere around it. Not sure if that influences the result at all.

 

[Leotron]

The imaginary sphere has the same material as the sphere around it. Not sure if that influences the result at all.

But when I was doing another problem which deals with calculating the self inductance of a toroid with a small gap, whose permeability is μ for the iron part and μ0 for the gap, the answer uses μ0 to calculate the B field in the gap (used "imaginary wire loop")...And the answer is definitely different using ε/ε0 in the sphere case. Still not sure why. Maybe the answer of that problem is wrong...

 

[mfb]

For the toroid magnet, you don't remove something, you treat it as circle with two different materials the whole time. And you don't have a charge distribution as you have here.

 

[Leotron]

For the toroid magnet, you don't remove something, you treat it as circle with two different materials the whole time. And you don't have a charge distribution as you have here.

Woo that makes sense...Thanks!