Calculation of magnetic/electric fields

• daudaudaudau
In summary, the conversation discusses equations for finding the B-field inside a superconductor or ordinary conductor, the Meisner effect, the London equation, and the charge distribution inside an electrical conductor. The problem of calculating the magnetic field inside a conductor with a given constant field outside is also mentioned, along with the concept of magnetization and references for further reading.
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?

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??

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.

Naty1 said:
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?

daudaudaudau said:
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.

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.

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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

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Naty1 said:
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
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.

Jackson, Classical Electrodynamics
Stratton, Electromagnetic Theory

What is the difference between magnetic and electric fields?

Magnetic fields are created by moving electric charges, while electric fields are created by stationary electric charges.

How are magnetic fields calculated?

Magnetic fields can be calculated using the formula B = μ0*I/2πr, where B is the magnetic field strength, μ0 is the permeability of free space, I is the current, and r is the distance from the current.

What is the significance of the direction of a magnetic field?

The direction of a magnetic field is important because it determines the force and motion of charged particles within the field.

How does the strength of an electric field change with distance?

The strength of an electric field decreases with distance from the source, following the inverse square law. This means that as the distance from the source doubles, the electric field strength decreases by a factor of four.

Can electric and magnetic fields interact with each other?

Yes, electric and magnetic fields can interact with each other. This is known as electromagnetic induction and is the basis for many technologies, such as generators and motors.

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