Magnetic field around a wire ?

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Discussion Overview

The discussion revolves around the magnetic field generated by an infinite current-carrying wire, particularly when wrapped with dielectric materials and superconductors. Participants explore the application of Ampere's Law and the implications of boundary conditions in different scenarios, including the presence of superconductors and their effects on the magnetic field.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests using Ampere's Law to find the magnetic field outside a dielectric material surrounding an infinite wire, while proposing to use the H field inside the dielectric and relate it to B using magnetic susceptibility.
  • Another participant proposes that the standard magnetic field equation for an infinite wire can be used, noting that the tangential H field must remain continuous across boundaries.
  • A question is raised about the scenario of wrapping a superconductor around the wire, where it is suggested that the B field outside the superconductor would be zero and queries whether Ampere's Law would still apply.
  • A later reply indicates that the presence of surface currents in a superconductor complicates the boundary conditions, and mentions the Meissner effect, which requires additional considerations regarding Maxwell's Equations.
  • It is proposed that deriving differential equations for the static magnetic field in the presence of a superconductor and dielectric materials could provide clarity on the situation.

Areas of Agreement / Disagreement

Participants express differing views on the application of Ampere's Law in the presence of superconductors and the implications of boundary conditions, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants acknowledge the complexity introduced by superconductors and the need for careful consideration of boundary conditions and the effects of surface currents, but do not reach a consensus on the implications for Ampere's Law.

cragar
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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 let's say I want to find the B field when I am outside the dielectric, Could I just use amperes law . And when I am inside the dielectric I would use the H field and then get B by using the magnetic susceptibility of the material . Is this correct .
 
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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 continuous H fields. So this is situation where Ampere's Law will give the correct result.
 
Thanks for your response , ok what if i wrapped a superconductor around the wire and cooled it with liquid nitrogen .
And let's say the superconductor goes 1 cm around the wire . Outside the superconductor the B field would be zero , Would amperes law still work?
 
cragar said:
Thanks for your response , ok what if i wrapped a superconductor around the wire and cooled it with liquid nitrogen .
And let's say the superconductor goes 1 cm around the wire . Outside the superconductor the B field would be zero , Would amperes law still work?

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.
 
wow , thanks for your answer .
 

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