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A cylindrical capacitor (two circular metal plates in vacuum) creates an electric field profile. As a thought experiment, can I set up currents to make a magnetic field which completely matches this profile inside the cylinder?

The currents can be placed anywhere (e.g, outside the cylinder with 3D complexity), but I'd prefer to have divergent-free currents, so that all fields are static. To be clear, magnetic monopoles are not allowed (the reverse problem of coverting a solenoid's magnetic field is easily solved by placing electric monopoles on the cylindrical boundary).

field

bobuhito
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  • This should be possible without any currents inside the capacitor, as long as we're talking about the static $\vec E$-field. What you're really asking is, "Please calculate this $\vec E$-field, find any nice continuation of the field lines inside out to infinity, and calculate the resulting $\nabla \times E.$" For a $\vec B$-field that will be a current density $\vec J.$ Within the cylinder this will be $0$ because $\partial_t B = 0$ and so the remainder is entirely due to currents outside of the space via the way we choose this "nice continuation." – CR Drost Jan 31 '17 at 00:13
  • My gut says that the best way to go with this is to find some sort of analytical expression for the field of a uniformly charged disk of charge, figure out a current pattern for that, and then replicate it twice symmetrically; but googling around suggests that nobody wants to solve for this analytical expression in practice, meaning that it probably involves some sort of special functions or something. – CR Drost Jan 31 '17 at 00:15
  • Well, I have my doubts when magnetic monopoles are not allowed...I'm really more interested in an existence/nonexistence proof than finding the exact analytic solution. – bobuhito Jan 31 '17 at 03:02

2 Answers2

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This can be done by finding the charge density on the plates and then mimicking it with solenoids from infinity. This probably therefore amounts to circular currents above the top plate (extending infinitely above the top plate) and the mirrored currents below the bottom plate.

I was really hoping for a solution which does not require infinite volume, so await more answers or a proof that infinite volume is required.

Community
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bobuhito
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  • After further thought, it seems I should be able to do this in finite volume by wrapping the solenoids around from one plate to the other (instead of going to infinity), but I hate to disrupt the rotational symmetry... – bobuhito Feb 02 '17 at 19:34
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No, you can't do this. The explanation is that the electric field lines will "stop" on the inner boundary; magnetic field lines have to be continuous, so somewhere inside the cylinder these lines would be traveling "up and out" to form closed lines of flux. This is the approximate distribution of field lines in the capacitor:

enter image description here

For a magnet, those arrows don't stop at the -ve charged plate.

Floris
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  • Sorry for the confusion, but I meant for the original capacitor to be simpler: It is just two metal plates (in one plane) separated to form a capacitor, just like the most basic textbook capacitor. – bobuhito Jan 30 '17 at 22:48
  • It's still not the same (in the sense that the field lines cannot point the same directions) because E-field lines DO dead-end on charges. B-field lines never dead-end. Answer might be different if only field intensity is considered (and depending on geometry: the disk-capacitor and cylinder-capacitor are very different cases). – Whit3rd Jan 30 '17 at 23:04
  • @Whit3rd I understand that, but I am only looking for it to be the same inside the cylinder, so was hoping the dead-end differences could be moved onto or outside of the cylinder. – bobuhito Jan 30 '17 at 23:13
  • How can it be both a cylinder (as I tried to draw) and a parallel plate capacitor (which you state in your first comment)? The diagram in your updated question - is that your idea of a the capacitor (two circular parallel plates)? Do you care about the field outside the homogenous field region? Are you trying to match the "bowing" of field lines towards the periphery? – Floris Jan 30 '17 at 23:34
  • @Floris The diagram in my updated question (which is a parallel plate capacitor) is correct. Yes, I am trying to match the "bowing" you mention. – bobuhito Jan 31 '17 at 00:00