Laplace in electromagnetics(voltages are different in conductor?)

[baby_1]

Hello
as you see this example and solution

if we assume the R of conductor sphere is 5m and check voltage in different z we obtain(for example z=1m and z=2)

and as we know in conductor we doesn't have voltage differences so this equation should be the same

C=0
but voltage aren't the same

why in the conductor sphere voltages aren't the same? or i do mistake to understand this example

 

[Naz93]

I think you're misunderstanding the example. The form of V -> -Ez + C only applies outside the sphere. The potential inside the sphere takes a different form. Since the sphere is uncharged, you can work out the potential inside it very simply - like you say, the potential should be the same at all points inside the sphere. And since it has been set to 0 at the boundary in the problem, this means that it has to be 0 everywhere inside the sphere.

 

[rude man]

This problem belongs in the Advanced Physics forum IMO.

The solution to Laplace's equation del²V = 0 for this case (azimuthal symmetry) involves Legendre polynomials.

You can obtain a closed-form expression for the potential everywhere outside the sphere including just outside its surface with the given boundary conditions and using just the 1st order polynomial in spherical coordinate θ.

As naz93 said, the potential inside the sphere is everywhere the same (call it zero). This is a very elementary fact of electrostatics. Any body with finite conductivity will have zero E field inside it, thus the potential does not vary inside of it.

 

[Fernbauer]

The problem can be solved without full Legendre polynomials. As a hint, consider an alternate problem of the same uniform electric background field but with a lone +Z dipole at the origin. Find the potential everywhere in space. Is there some dipole magnitude such that the net potential on the unit sphere is a constant? What does that tell you about your sphere problem?

 

[rude man]

The problem can be solved without full Legendre polynomials. As a hint, consider an alternate problem of the same uniform electric background field but with a lone +Z dipole at the origin. Find the potential everywhere in space. Is there some dipole magnitude such that the net potential on the unit sphere is a constant? What does that tell you about your sphere problem?

Dipole? Quadrupole I can see.