Voltage across rod in time varying magnetic field

Question

If a slim conductor of some length $l$ and diameter $d\ll l$ is placed in a magnetic field $B$, and the field is changed by $\frac {dB}{dt}$, what (if any) is the voltage $V$ induced across the ends of the conductor?

In my case of interest, the slim conductor is a wire, fixed in space, that is victim of interference by an adjacent current, where the $\frac {dB}{dt}$ is caused by $I\ \sin(\omega t)$ in a source wire.

I am particularly interested in calculating a specific case (given $I_0$, $\omega$, and $r$ the distance between the two wires), as well as the fundamental connections to Maxwell's laws, probably the Maxwell- Faraday equation.

I am familiar with Lenz's law , but in my case of interest there is no return path or "ground plane", and so the victim wire has no current loop, or EMF loop. I can't form a curl integral, and no area is determined, and thus no time varying flux. Nevertheless, I would expect the above "wire rod" case to be the Maxwellian foundation of, or at least a step towards the Lenz "loop around flux" case. (Or perhaps I am terribly on the wrong foot here.)

The closest I come to this problem is by the Lorentz force , as it also involves a rod, and it involves a EMF on a charge in motion in a magnetic field. In contrast, my question centers around a time-varying magnetic field, without motion.

To be clear, the rod is fixed in space, and I am interested in the voltage calculation, not the motion or forces.

(Image from https://www.aplusphysics.com/courses/regents/electricity/images/InductionProblem.png)

A related answer is provided in Induced electric field from homogeneous magnetic field but this leads to a inhomogeneous current distribution and E-field, and I am not sure what to make of it in the case of a rod.

Answer 1

When the magnetic field is changing, there is no notion of "voltage" only of EMF. If you were to couple the two ends of your rod to a voltmeter, its reading depends on the time rate of change of the magnetic flux in the loop composed of the rod and wires connecting it to the voltmeter. Different paths for the wires will give different "voltage" readings, so the "voltage" is not meaningful. The voltmeter records the EMF which is the force trying to drive a current through the the meter.

Answer 2

If the (time-varying) field surrounds the wire equally to the right and left of the wire, symmetry surely dictates that the voltage is zero: neither end of the rod gains a charge.

Try formulating a 'hand rule' to predict the direction of the induced emf. Point the index finger (say) of either hand in the direction of increasing magnetic field. Second finger or thumb can still point in any direction at right angles to the index finger; no information about the direction of an induced emf can be obtained. The situation is too symmetrical – and too symmetrical for there to BE an emf!

If the field arises from a changing current in a parallel wire, the symmetry is broken and there is an induced voltage in the 'first' wire. For details, look up 'Mutual inductance of parallel wires'.

Answer 3

If a slim conductor of some length l and diameter d<<l is placed in a magnetic field B, and the field is changed by dB/dt, what (if any) is the voltage V induced across the ends of the conductor?

When $B$ is varying, a varying $E$-field (call it external) also appears along the wire. Therefore, I think, if your induced voltage produces an electric field inside the wire (call it internal) which is in the direction of the external varying $E$-field, the wire or rod accelerates along the $E$-fields, and in the meanwhile, it rotates about its center of mass because the internal $E$-field, due to your induced voltage, has accumulated the positive and negative charges, respectively, at each end of the rod, and thus the motion of these charges in the $B$-field produce a torque on the rod due to the Lorentz forces exerted on the rod's ends in the opposite directions.