Magnetic field around a wire ?

cragar

Lets say we have an infinite current carrying wire and it is wrapped with a dielectric materiel.
Lets say the dielectric is 1 cm in diameter around the wire. But lets say I want to find the B field when Im outside the dielectric, Could I just use amperes law . And when im inside the dielectric I would use the H field and then get B by using the magnetic susceptibility of the material . Is this correct .

Born2bwire

Off the top of my head I think you can just use the standard magnetic field equation for an infinite wire. The boundary conditions require that the tangential H field be continuous. Since the B field will be described as

[tex]\mathbf{B}(\rho) = \frac{\mu I}{2\pi\rho} \hat{\phi} [/tex]

Since the fields are tangential to the inhomogeneity they will not be affected since they will automatically have continous H fields. So this is situation where Ampere's Law will give the correct result.

cragar

Thanks for your response , ok what if i wrapped a superconductor around the wire and cooled it with liquid nitrogen .
And lets say the superconductor goes 1 cm around the wire . Outside the superconductor the B field would be zero , Would amperes law still work?

Born2bwire

That one gets tricky. First, there have to be currents excited over the surface of the conductor to expel the applied field. This changes the boundary conditions because now the tangential H fields is related to the surface currents. In addition, it is my recollection that the Meissner effect requires extra stipulations on Maxwell's Equations. For example, if we were to solve for a static magnetic field applied in the presence of a perfect electrical conductor, then the solution only states that the field inside the PEC has to be constant. What this constant field is can be zero or the applied field depending upon how you look at the conditions of the problem I believe. However, with a superconductor there are added conditions that remove this ambiguity and ensure that the result is no magnetic field.

Perhaps the best thing to do would be to find the differential equations describing a static magnetic field excited by a static current. Then derive a list of boundary equations dictated by your superconductor and dielectrics and use these in conjunction with your differential equation and solve. This would be similar to, for example, solving the Poisson equation to find the electrostatic field.

cragar

wow , thanks for your answer .