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

• I
• swampwiz
In summary, the conversation discusses the problem of a wire-loop moving relative to a bar magnet. The case of the wire-loop being fixed results in a changing magnetic flux through the loop, causing a net electrical field according to Faraday's law. On the other hand, the case of the bar magnet being fixed suggests that the lateral component of the magnetic field, rather than the component in the direction of motion, is responsible for the force on charges in the wire-loop. This may be confusing for those accustomed to thinking about magnetic flux in the centerline direction. However, since the wire-loop's motion is also in the centerline direction, the cross-product of parallel vectors is 0, resulting in a force of 0 on the charges.
swampwiz
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?

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.

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