I have had a problem understanding what the product of charge q with vector potential
A (bold denotes a vector), (q*
A), really means. It is regarded as a form of hidden momentum in the charge q and any change in momentum results in a force
F. Hence
F=-q*d
A/dt, then if we recognize
F as due to an effective induced
E field we get
E=-d
A/dt. That is fine when
A is changing with time. But when you have a static but non-uniform
A field, when the charge q moves through it at some velocity it sees a time changing
A field so should endure a force due to that non-uniformity. For non-uniformity such that the charge sees a magnetic field we have well known laws dealing with forces, but we can also have non-uniformity that is free of magnetic fields and that is the interest here. If you take the simple case of an
A field that points along (say) the x axis everywhere along that axis but its magnitude changes then a charge moving linearly along that axis sees that change and you can deduce that
Fx=-q*(d
Ax/dx)*
v where
v is the velocity along the x axis. That is OK but if we want to make use of that the charge cannot continue along that axis indefinitely. We need charge flowing in closed circuits and that means a change of velocity direction as charge moves around the circuit. You can no longer just use the component of
A along the velocity direction (the tangential component) and use it to evaluate the change in
A seen by the charge as it gives the wrong answer. This becomes clear if you consider charge moving in a circular orbit within a uniform
A field, the tangential component follows a sine wave that suggests a sinusoidal force, but in fact the hidden momentum remains constant, there is no change hence no force. That realization that the moving charge can change direction within an
A field yet endure zero force takes on a serious meaning if we have a hairpin circuit where the conductor is folded back on itself. Over the tiny dimensions of the fold the
A field is virtually uniform even if it changes value away from the fold.
Thus, we can have charge moving within a conductor that sees a 180-degree change in A field direction but without seeing any force. That is a form of regauging. If the charge moving towards the fold is seeing an increase in
A that is a positive d
A/dt yielding a negative force. If at the point of maximum
A it reaches the fold, it then finds itself seeing a negative
A and as it continues is reverse journey that negative
A reduces in value.
That reverse journey is still seeing a positive dA/dt yielding a negative force. I am not aware that this rectifying aspect of circuits within an
A field has been noted before, and it leads to closed circuits enduring an induced voltage, something that current science {and F6
} says is impossible.
The interesting aspect of the induced voltage is its magnitude being proportional to the velocity of the charge. For a given size of conductor the velocity is related to the current and the volume density of the charge carriers, hence the induction appears as a resistance value that can be either positive or negative depending on the direction of the current. Of course it is the induced negative resistance that is of interest here. For copper conductors within a practical embodiment the negative resistance is likely to be a tiny fraction of its normal positive resistance, but with superconductors that have a much smaller volume density of carriers (Cooper pairs) the velocities are much greater and there is zero positive resistance. Thus, it is possible to conceive of a system where current builds up from circuit noise to some high value within a multi-turn closed "coil" of hairpins, the coil is opened and connected to a load so that it discharges its energy, then it is closed again and the cycle repeats. Luckily I have experience of super-regeneration so I can work all this out.
Smudge
Author
Topic: Generator using a superconductor (Read 3745 times)
