How to Model a Magnet Falling Through a Solenoid

[kuruman]

Table of Contents
IntroductionThe motional emfA single ringFrom ring to solenoidThe emf profileTerm correctionsA relation between...

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[Delta2]

Great article for an experiment that is relatively simple to setup and perform, however it has a not so simple detailed explanation using the laws of classical electromagnetism.

 

[kuruman]

Great article for an experiment that is relatively simple to setup and perform, however it has a not so simple detailed explanation using the laws of classical electromagnetism.

When you say "not so simple", what specifically are you referring to? The math or the physics? Both, I think, are appropriate to the undergraduate intermediate level.

 

[Delta2]

When you say "not so simple", what specifically are you referring to? The math or the physics? Both, I think, are appropriate to the undergraduate intermediate level.

Yes I agree they are appropriate for undergraduate level, it's just that they are not so simple as for e.g. the emf of a rotating ring in a uniform B. The expressions look a bit complex, main responsible for this is the form of the B- field from a point dipole.

 

[Charles Link]

Very good article @kuruman :). Besides the thread by @billyt_ mentioned in the above article, it may also be of interest to see posts 122-123 and 138-139 of https://www.physicsforums.com/threads/calculating-magnetic-field-strength-of-a-magnet.1005917/page-4
@hutchphd and @bob012345 had very good inputs in helping to solve the problem of the EMF of a magnet moving through a ring.

 

[kuruman]

Very good article @kuruman :). Besides the thread by @billyt_ mentioned in the above article, it may also be of interest to see posts 122-123 and 138-139 of https://www.physicsforums.com/threads/calculating-magnetic-field-strength-of-a-magnet.1005917/page-4
@hutchphd and @bob012345 had very good inputs in helping to solve the problem of the EMF of a magnet moving through a ring.

Thank you for liking the article and for pointing out this other thread. Part of my motivation for this article was to consolidate in one place what can be said about magnets falling through a single ring and a solenoid assuming the point dipole approximation. My hope is that it can serve as a reference for future questions and as a starting point for more complicated magnet points.

I should mention (advertise?) that I am now putting the finishing touches on a companion article that is intended to serve the same purpose but for a magnet falling through a solid conducting pipe. It should be ready in a few days.

 

[vanhees71]

You can also complete the article by actually describe the motion of the magnet itself. It's nicely described, e.g., in

https://doi.org/10.1119/1.2203645

He is using a different model for the magnet, i.e., a cylinder of homogeneous magnetization, which has been demonstrated to be a quantitatively better model for the experiment with a real bar magnet than the dipole approximation:

https://doi.org/10.1119/1.4864278

 

[Charles Link]

With this topic, (of the permanent magnet), perhaps it would even be useful for some readers to read come of the fundamentals. See
https://www.physicsforums.com/threa...by-magnetic-surface-currents-comments.900528/
and
https://www.physicsforums.com/threads/a-magnetostatics-problem-of-interest-2.971045/

When I was in college, (many years ago), we were taught the pole method, but it is important to connect that method to the magnetic surface current method, because otherwise, it is a lot of incomplete handwaving, where you have static fictitious charges creating a magnetic field. Fortunately, as it turns out, the pole method follows from the magnetic surface current method, and the two get the exact same answer for the magnetic field.

( I don't want to steer the reader away from the problem of the permanent magnet falling through the solenoid, but it is kind of important to have complete physics for the permanent magnet, in order to work this problem).

 

[kuruman]

He is using a different model for the magnet, i.e., a cylinder of homogeneous magnetization, which has been demonstrated to be a quantitatively better model for the experiment with a real bar magnet than the dipole approximation:

https://doi.org/10.1119/1.4864278

Yes, I agree that the permanently magnetized disk model produces a better quantitative model if one has the data on one hand and the sophisticated analysis on the other. Like I said, the point dipole as a model is as useful and as accurate as the point mass is in mechanics.

My goal with these articles is modest: to show, at the intermediate undergraduate level, that the point dipole model goes a long way towards understanding what is physically going on with a magnet falling through a solenoid and a pipe. It's not perfect, but it does a descent job and any refinements to the model for a magnet are not going to change the basic features of the point dipole model. A secondary goal is to gather these ideas all in one place as an easily accessible reference for PF users who have questions related to this material.

 

[Charles Link]

See https://www.physicsforums.com/threads/drop-height-of-a-magnet-vs-induced-emf-in-a-solenoid.1014918/ (thread by @billyt_ )
posts 25 and 39. I think the transition from point magnetic dipole to an extended permanent magnet is fairly straightforward, because we are taking $d \Phi /dt$ to compute the EMF, and thereby the integral ($d \Phi/dt$ rather than $\Phi$) over the length of the solenoid becomes routine. The magnetic dipole is also a very good calculation, but if we can do a finite length permanent magnet in closed form, that is even somewhat better. It's a little bit of extra work, but writing out the terms for the case of the extended permanent magnet would make the reference more complete.

Edit: On second thought, it is perhaps worth mentioning, but really not worth the trouble of displaying all the terms, since it is too lengthy.

 

[vanhees71]

But you can make your Insight easily complete by using your point-dipole model by calculating the equation of motion for that point dipole, falling through the pipe. See the quoted papers above.