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Maxwell asserted that a virtual current. the displacement current, which was proportional to the charging field between the plates was sufficient to produce a magnetic field of the same orientation as the magnetic field surrounding the charging conductors connected to the capacitor plates. The fact that no charging electrons needed to be present was the basis of his theory of electromagnetism since a changing electric field alone could produce a magnetic field.

Do the mobile electrons in the connecting wires lose their ability to generate a magnetic field when they flow onto the capacitor plates? The left hand rule for negative charge gives the field above the charging wire as entering into the x y plane and its return exiting below that conductor, and is consistent with the circular field surrounding the wire. When electrons flow onto the upper plate the same rule gives the field on the left of the plate to be entering the xy plane and its return exiting on the right side of the plate.

When the electron current flows onto the bottom plate the rule gives the magnetic field as exiting the x y plane on the left side of the plate but entering the x y plane on the right of the plate.

For every electron that arrives on the left plate another electron leaves the right plate, driven by the attraction of the battery and the Coulomb repulsion of the left plate. For the electrons leaving the top right plate the rule gives the direction of the magnetic field produced on the left side of the top right plate as exiting on the left side and entering on the right side of the plate which means that between the top plates the field is exiting. The rule then gives the field as entering the x y plane between the bottom plates. Thus there is a magnetic field between the plates which is opposite in circulation to the fields surrounding the charging wires.

This contradicts the displacement current notion that the magnetic field between the plates has the same direction of circulation as that of the charging wires and is solely a consequence of the changing electric field.

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I believe you’re asking “What is the magnetic field due to charges flowing radially from the center of a plate to its edges?”

The answer is zero.

You describe the field from one radial bit of charge motion. But now imagine the field from the bits on either side: because the field wraps around the charge motion, these will be off opposite direction to the original and cancel it.

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  • $\begingroup$ Offhand not a bad answer but Ampere's Force Law states that anti parallel currents repel ,not cancel and thus we define the ampere as 2x10-7 newtons if the conductors are separated by one meter. $\endgroup$ – Brobin Jul 30 '18 at 14:17
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    $\begingroup$ @Brobin True, but doesn’t effect this situation. If you look at the 3D shape of the capacitor, once current reaches the plates & spreads out there’s no way for it to generate a B field around the original direction. Only displacement current can do that. $\endgroup$ – Bob Jacobsen Jul 30 '18 at 15:15
  • $\begingroup$ There is no way to generate a mag field once the electron flow has ceased. The capacitor is charged. Grasp the "up" electron flow on the first plate with the left hand. Your fingers will curl around and exit the space between the plates. Grasp the "down" current with left hand with the forefinger in the direction of electron flow. The fingers will again curl around to the space between the plates. The magnetic field is TWICE the field around the input wire. Or use the right hand for the conventional current. The answer will be the same. $\endgroup$ – Brobin Jul 31 '18 at 12:10
  • $\begingroup$ @brobin. I think I’m not understanding your question. There’s also no displacement current once the electron flow is zero. Nowhere is there a contradiction with displacement current. $\endgroup$ – Bob Jacobsen Jul 31 '18 at 12:18
  • $\begingroup$ I think I am beginning to see your problem. Maxwell asserted that the displacement current had a magnetic field between the plates that had the same orientation as the charging current. As we have demonstrated the orientation (clockwise versus counterclockwise) is 180 degrees out of phase with the input currents field. $\endgroup$ – Brobin Jul 31 '18 at 15:39

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