How do I apply Maxwell's equations?

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Discussion Overview

The discussion revolves around the application of Maxwell's equations, particularly in calculating the electric field generated by a changing magnetic field across a surface. Participants explore theoretical and practical aspects of this problem, including the implications of different boundary conditions and the nature of the surfaces involved.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how to calculate the electric field at a surface when a magnetic field perpendicular to it is changed at a constant speed, noting the ambiguity in the integral ∫E⋅ds.
  • Another participant suggests that under certain conditions, the electric field E could be expressed as k/s, referencing the divergence of E being zero in regions without charge.
  • Some participants raise the distinction between mathematical and physical surfaces, indicating that physical materials may have charges and could require boundary conditions to solve Laplace's equations.
  • One participant expresses confusion about applying Gauss's law or Laplace's equations when the curl of E is not zero, emphasizing the complexity of the electric field configuration without charges present.
  • Another participant points out that the problem is under-specified, mentioning scenarios like a Helmholtz coil creating a varying magnetic field and the potential for overlapping static electric fields from external charges, which complicates the solution to Maxwell's equations.
  • It is suggested that if only the Helmholtz coil's effect is considered, symmetry could be used in conjunction with Faraday's law to derive the electric field.

Areas of Agreement / Disagreement

Participants express various viewpoints on the nature of the surfaces and the conditions under which Maxwell's equations apply. There is no consensus on how to approach the problem, with multiple competing views and unresolved questions regarding boundary conditions and the specifics of the electric field configuration.

Contextual Notes

The discussion highlights limitations in specifying boundary conditions and the assumptions regarding the nature of the surfaces involved. The complexity of the electric field due to the interplay of different fields and the lack of charge in certain regions are also noted as factors that complicate the analysis.

Lasha
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For example, if I have a magnetic field perpendicular to some surface and I change this magnetic field with constant speed, how do I calculate the Electric field at any point on this surface, since ∫E⋅ds=k, where k is some constant, could be done with many different vector fields.
 
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Or does E always equal k/s cause ∇⋅E=0 where there's no charge?
 
Are you talking about a mathematical surface or a physical surface? A physical material can have charge and may be made of dielectric,diamagnetic materials. For some cases, you need to apply some boundary conditions and then solve Laplace's equations. But for simple cases, you can often look at the symmetry of the problem and apply Gauss's law.
 
Khashishi said:
Are you talking about a mathematical surface or a physical surface? A physical material can have charge and may be made of dielectric,diamagnetic materials. For some cases, you need to apply some boundary conditions and then solve Laplace's equations. But for simple cases, you can often look at the symmetry of the problem and apply Gauss's law.
I don't get it how do I solve it with Gauss or Laplace when ∇×E≠0. I don't have a charge or even a region where electric field is made by a charge.I simply have sum of many circular vectors of E at any point on this surface.
 
There are many possible fields because the problem is under-specified. You could have a Helmholtz coil which creates a varying magnetic field in a space, and you could have charges outside the region of interest. The charges would produce a static E field which overlapped the E field due to the Helmholtz coil. Since this is a valid physical situation, it's a solution to Maxwell's equations. So you can have all sorts of different static E fields superimposed with of the Helmholtz coil field which are all solutions to Maxwell's equations. This is why you need to specify boundary conditions.

If you are looking for the solution which is just due to the Helmholtz coil itself (no static fields from charges outside the region of interest), then you can apply symmetry to Faraday's law to get E.
 

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