Calculation of magnetic/electric fields

[daudaudaudau]

Hi.

For a superconductor we have the nice equation $\nabla^2 \mathbf B=\frac{1}{\lambda^2}\mathbf B$. Using this equation we can find the B-field inside the superconductor if we have the boundary values. But what about an ordinary conductor(or dielectric) ? If I know what the field is outside the object, what is the equation I can solve to find the field inside?

 

[Naty1]

Your quoting the Meisner effect which shows the electromagnetic free energy in a superconductor is minimized when your equation is satisfied.

There is no corresponding "minimum" in everyday conductors...it's zero when there is no electric nor magnetic induction.

This "London equation" predicts the magnetic field in a superconductor decays exponentially from whatever value it possesses at the surface.

Or are you looking for the charge distribution inside an electrical conductor??

 

[ideasrule]

An ordinary conductor always has 0 electric field inside. For a dielectric, the charge inside depends on how the charge was deposited in the first place and can't be calculated from surface charge.

 

[daudaudaudau]

Your quoting the Meisner effect which shows the electromagnetic free energy in a superconductor is minimized when your equation is satisfied.

There is no corresponding "minimum" in everyday conductors...it's zero when there is no electric nor magnetic induction.

This "London equation" predicts the magnetic field in a superconductor decays exponentially from whatever value it possesses at the surface.

Or are you looking for the charge distribution inside an electrical conductor??

My problem arose because I wanted to calculate the magnetic field inside a conductor (not necessarily a perfect conductor) if there is a given constant field outside. This seems to be extremely easy for a superconductor, because we have that nice equation I quoted, but what about an ordinary conductor? How can I actually calculate the field?

 

[Born2bwire]

My problem arose because I wanted to calculate the magnetic field inside a conductor (not necessarily a perfect conductor) if there is a given constant field outside. This seems to be extremely easy for a superconductor, because we have that nice equation I quoted, but what about an ordinary conductor? How can I actually calculate the field?

My recollection is that it's a fairly poorly posed problem. Without the assumptions of the superconductor's properties, I recall that the magnetic field inside the conductor, due to an externally applied static magnetic field, must be constant. Assuming that the permeability is the same as vacuum inside the conductor then I believe that the magnetic field is unaffected by the presence of the conductor. But again I think that when assuming perfect electrical conductors that this is not a mathematically pleasant problem to define.

 

[marcusl]

If the material is non-magnetic (permeability mu=1) then the field penetrates freely as though through a vacuum. If not, you can calculate the magnetization M

$$\vec{M}=\chi\vec{H}=\vec{B}-\mu_0\vec{H}$$

where chi is the magnetic susceptibility. It is a little involved because of the so-called demagnetizing field, which depends on the shape of the object and direction of applied field.
EDIT: Corrected sign above.

 

[Naty1]

My problem arose because I wanted to calculate the magnetic field inside a conductor (not necessarily a perfect conductor) if there is a given constant field outside.

I wondered if this is what you are after...anyone have a reference that explains the phenomena a bit...

An ordinary conductor always has 0 electric field inside

for an ideal conductor...not a real world imperfect conductor...

I don't have the background to provide any concrete answer but I would think paramagnetism and diamagnetism of the material would be relevant:
http://en.wikipedia.org/wiki/Magnetic_permeability

 

[marcusl]

I wondered if this is what you are after...anyone have a reference that explains the phenomena a bit...

If $$\mu=1$$ then the field penetrates as in a vacuum. If not, see the equation I gave above for B inside.
For references, see
Undergrad level:
Reitz and Milford, Foundations of Electromagnetic Theory, has a very nice treatment of magnetization (I have the 1st edition, in case it matters).
Griffiths is likely to be good based on reputation, though I don't own a copy.

Advanced level:
Jackson, Classical Electrodynamics
Stratton, Electromagnetic Theory