# E-field generated by constant current in a circular loop

## Homework Statement

Discussions of the possibility of a tangential E-field external to a current-carrying conductor must include a voltage source and a return path. Here the problem is reduced to (we believe) the simplest possible geometry. Assume that a circular conducting loop has constant emf per unit length and constant resistance per unit length (finite conductivity), and therefore constant DC current. Assume that the loop has a non-zero conductor diameter and zero net charge. We can also assume that Rloop >> Rconductor. A conceptual model might be, for instance, the limiting case of a loop formed from an arbitrarily large number of arbitrarily small solar cells under constant illumination.

## Homework Equations

In considering the associated E-field, for any arbitrary reference point on the loop, the maximum voltage differential is across the diameter. This is equally true if the reference point is moved to the other end of the diameter, the difference being that the E-field now points in the opposite direction. This leads to the conclusion that there is no inward-directed (radial) E-field inside the loop, and by similar reasoning nor is there one outside of it. Inside the conductor, there is an E-field parallel to the axis and equal to current times resistance per unit length.

## The Attempt at a Solution

The question is whether there is any E-field exterior to the conductor, tangential to the surface, and extending beyond it? Some analyses say yes, but ignore the imputed infinite impedance of a DC E-field (see for instance http://www.dannex.se/theory/3.html and figure 5). If the free-standing tangential E-field is real, it would exert motive force on a test charge, which could be the basis for power transfer to a parallel, independent conductor, e.g. a "DC transformer" sans coupling by magnetic means. This effect is not demonstrated to exist. Even superconducting DC transformers are based on magnetic coupling. Is the purported tangential field nugatory here or have I missed something fundamental?

Delta2
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In my opinion yes there could be external tangential E-field, depends on the exact geometry of the DC circuit.

Regarding your argument why we cant observe a "DC transformer effect" , that is because the hypothetical parallel independent conductor will initially get some minimal power from the tangential E-field, but will reach electrostatic equilibrium very soon, and hence there would be absolutely no current and no E-field inside that conductor. The electrostatic induction cant make a DC transformer work because eventually we ll reach electrostatic equilibrium, need magnetic /electromagnetic induction for a transformer to work.

That is not an entirely satisfactory answer. All you are saying is that an open circuit wire has zero current once the stray capacitance has charged up. However, if the tangential field is real, the voltage induced in a wire parallel to it should be both measurable and able to drive continuous closed-loop current through a load resistor. For the stated circular geometry, the external field supposiiton posits a free standing, closed loop E-field similar to that for AC drive but at DC. The AC version is the basis of ordinary transformers. When dB/dt= 0, as is the case here, there is no induced closed loop E-field by that mechanism. If the tangential DC field is real, then how does it arise? I think we are agreed that there are no (non-superconducting) DC transformers. What I am saying is that, for the reasons stated, this is contrary to the tangential DC E-field hypothesis.

Delta2
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However, if the tangential field is real, the voltage induced in a wire parallel to it should be both measurable and able to drive continuous closed-loop current through a load resistor.
No, I don't agree because the whole closed loop with the load resistor and the parallel conductor will also reach electrostatic equilibrium. A closed loop is just a specific type of conducting structure. Any conducting structure under the influence of an external constant (with respect to time) E-Field will reach electrostatic equilibrium. Unless ofcourse there is some separate emf source inside the conducting structure.

EDIT: IF you view the parallel conductor as an EMF source, you do very wrong in your view. It is rather some sort of capacitor that will discharge very soon and the whole closed loop will reach electrostatic equilibrium. An EMF source has internally some chemical or electromagnetic mechanism that creates an EMF E-field that counters the electrostatic field from the poles, so the total E-Field is zero inside an EMF source.

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So you are saying that a free standing, closed loop E-field is incapable of inducing other than electrostatic equilibrium? Speaking as a magnetics designer, I'm not sure you follow the transformer analogy. Closed loop E-fields most definitely induce voltage in and drive current through wires parallel to the field. This should also happen at DC, unless of course there is no equivalent field at zero frequency. If you assert electrostatic equilibrium for the given circular geometry, exactly where are the equilibrium charges piling up?

Delta2
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We are not talking about closed loop E-fields here, those fields are non conservative and can arise only in time varying situations. In a DC circuit we can only have static, conservative E-fields, inside and outside of the circuit.
Are you saying that a circular loop of conducting wire can not reach electrostatic equilibrium under the influence of an external static conservative E-field? I am sorry but i think you are wrong. The equilibrium charges distribute through the circumference in such a way as to cancel the external static E-field inside the circumference.

We ARE talking about a closed loop E-field. If there is a tangential E-field outside of the conductor, it is required by the circular geometry of the loop. I agree totally that those arise only in time-varying situations. That's the whole point - you cannot have one at DC! Again, electrostatic equilibrium between where and where? From the beginning I pointed out the non-existence of a radial field. What is the direction of the resultant E-field in your model?

Delta2
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We ARE talking about a closed loop E-field. If there is a tangential E-field outside of the conductor, it is required by the circular geometry of the loop. I agree totally that those arise only in time-varying situations. That's the whole point - you cannot have one at DC!
I see your point now, but still I disagree it wont be a perfectly closed loop E-Field. It will be a conservative static E-Field that is mimicking a non conservative closed loop E-Field in the part of the circular loop that it is outside the EMF source. At the points of the circular loop where the emf source is located the electric field will reverse in direction , so inside the EMF source will have counterclockwise direction (if outside the emf source had clockwise direction), so it wont be a closed loop E-Field afterall.

Again, electrostatic equilibrium between where and where? From the beginning I pointed out the non-existence of a radial field. What is the direction of the resultant E-field in your model?
In my opinion there would be radial E-Field as well but anyway I don't want to argue about it, if we take a circular circuit with an EMF source, and outside of it we put a concentric conducting circular loop of wire but without an EMF source, then i still claim this concentric external wire will reach electrostatic equilibrium.

Delta2
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I now read your OP more carefully, its the way your setup from start and the assumptions you make that create a closed loop non time varying E-field. This type of field according to Maxwell's equations, simply cannot exist. That is, its curl would be non zero (we can arrive at this conclusion if we use stokes theorem in that closed loop), but its time derivative would be zero, and these two things are not allowed by Maxwell's equations.

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Now we fully agree. By extension, there also cannot be a current-driven DC E-field parallel to an infinitely long straight wire, which is just a loop that has reached the zero curvature limit. In a circuit with discrete voltage sources and folded current paths, there will always be E-fields associated with the voltage gradient along the current path, but these E-fields truly are electrostatic rather than current driven. In other words, if there was a knob that could somehow vary the conductor resistivity, the current could be adjusted without any corresponding alteration of voltage distribution. Electrostatic contributions only serve to confound the fundamental question of whether DC current flow creates a tangential E-field outside a resistive conductor. Fortunately, the electrostatic contribution is suppressible through use of simplified geometry, and the resulting answer is no. Have I stated this fairly?

Delta2
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I agree that the E-fields outside of a DC circuit are not current-driven (equivalently not due to the vector potential) but due to the scalar potential and the surface charge densities. So yes we can say that they are electrostatic and not current-driven.

But still I insist there can be electrostatic tangential e-fields outside a DC circular circuit with a DC EMF source (not of course a DC circular circuit with an E-field as you setup in your OP, that setup you do at the OP is not a valid DC setup).

We seem to have come full circle. Here I return to the question in my first reply: "If the tangential DC field is real, then how does it arise?" If it is real, surely it will impart motive force on a test charge. For instance, DC electrostatic fields are notorious for dust collection via this mechanism. Surely this mechanism is independent of the means of origin of the DC E-field. You say that "the OP is not a valid DC setup" despite provision of a plausible means to approximate it. In what way is it not valid? And if there is such a thing as a tangential E-field, what experiment could be used to verify that? Could we pass DC current through the center conductor of a coaxial cable and look for an induced voltage on the shield?

Delta2
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The tangential DC field can arise from the surface charge densities (as we agreed, it will NOT be current-driven, current driven E-fields come only from time varying situations) that will be induced in the circular circuit by a typical DC EMF source. BUT as I explain in post #8, the field that a typical DC EMF source setups is not a perfectly closed loop E-Field.
The field you setup at post #1 corresponds to a solution in Maxwell's equations in vacuum, with a magnetic field that is homogeneous everywhere in space and linearly time varying, and the resultant E-Field will be concentric constant in time, closed loops. But we simply just cant have such a magnetic field be produced either by permanent magnets or by current densities, it is just a theoretical solution to Maxwell's Equations.