I The vector math of relative motion of wire-loop & bar magnet

AI Thread Summary
The discussion centers on the vector math involved in the relative motion between a wire-loop and a bar magnet. It emphasizes that the changing magnetic flux through the wire-loop, as described by Faraday's law, generates an electric field due to the lateral component of the magnetic field, not the component aligned with the motion. The force on the charges within the wire-loop arises from this lateral magnetic field, as the motion is perpendicular to the magnetic field direction. This clarification is crucial for understanding the mechanics of electromagnetic induction in this scenario. Overall, the analysis presented is deemed accurate and highlights the importance of distinguishing between different components of the magnetic field.
swampwiz
Messages
567
Reaction score
83
I was watching this video about how the problem of a wire-loop moving relative to a bar magnet:



The case of presuming that the wire-loop is fixed seems to be that the magnetic flux (along the surface normal to the direction of the centerline - call it C) through the wire-loop is changing in time, thus causing there to be a net electrical field along the wire-loop, as per Faraday's law (or Maxwell's 3rd law). However, the case of presuming that the bar magnet is fixed seems to be that it is not the component of the magnetic field in the direction of the motion, but rather the component of the magnetic field in the direction going laterally away from the centerline of the magnet (call it R), such that charges of both sign-types are moving with a velocity in C, thus imparting a force (let's presume that the right-hand rule is C x R = T ) that is in the T direction, but in the direction as per the sign-type of charge, thus generating an electrical field along the wire; I would presume that the positive charges, the nuclei, resist the force, and that this is imparted back to the magnet (it would cancel out since it would be from a loop), but the negative charges, the electrons, get pushed through the wire loop, which is equivalent to there being an electric field in the wire.

I think the lecturer was not careful in explaining that it is the component of the magnetic field in the lateral direction, and someone who is used to thinking about magnetic flux through a wire-loop as the component in the centerline direction could very well think that it is this component causing the force - but that cannot be since the motion of the wire-loop itself is in the centerline direction, and since the cross-product of parallel vectors is 0, the force on the charges would be 0.

Is this accurate?
 
Physics news on Phys.org


Yes, your understanding of the vector math of relative motion between a wire-loop and a bar magnet is accurate. The key concept to understand is that the force on the charges in the wire-loop is not caused by the component of the magnetic field in the direction of motion, but rather by the component of the magnetic field in the lateral direction. This is because the motion of the wire-loop is perpendicular to the direction of the magnetic field, so the cross-product of these vectors is not zero, resulting in a non-zero force on the charges. It is important to clarify this point, as it may be confusing for someone who is used to thinking about magnetic flux through a wire-loop in terms of the component in the centerline direction. Overall, your analysis of the relative motion between the wire-loop and bar magnet is correct.
 
This is from Griffiths' Electrodynamics, 3rd edition, page 352. I am trying to calculate the divergence of the Maxwell stress tensor. The tensor is given as ##T_{ij} =\epsilon_0 (E_iE_j-\frac 1 2 \delta_{ij} E^2)+\frac 1 {\mu_0}(B_iB_j-\frac 1 2 \delta_{ij} B^2)##. To make things easier, I just want to focus on the part with the electrical field, i.e. I want to find the divergence of ##E_{ij}=E_iE_j-\frac 1 2 \delta_{ij}E^2##. In matrix form, this tensor should look like this...
Thread 'Applying the Gauss (1835) formula for force between 2 parallel DC currents'
Please can anyone either:- (1) point me to a derivation of the perpendicular force (Fy) between two very long parallel wires carrying steady currents utilising the formula of Gauss for the force F along the line r between 2 charges? Or alternatively (2) point out where I have gone wrong in my method? I am having problems with calculating the direction and magnitude of the force as expected from modern (Biot-Savart-Maxwell-Lorentz) formula. Here is my method and results so far:- This...
Back
Top