2
$\begingroup$

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.

enter image description here

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

$\endgroup$
0
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Thank you for your answer, Mohammad. Maybe I should clarify that the rod is fixed in space, and I am interested in the voltage, not the motion or forces. You write "has accumulated the positive and negative charges" which is the question. Has it? And if so, how is it calculated? $\endgroup$ – P2000 Aug 3 '20 at 15:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.