Lenz' Law copper tube thickness question

[Tesla1899]

For the classic Lenz law demonstration using a neo. magnet dropped down a copper pipe, is the magnetic force generated correlated to the thickness of the copper pipe? Ergo, will the magnet go slower down a thick copper pipe vs. a standard size plumber pipe?

 

[Delfador]

The velocity of the magnet down the pipe is inversely proportional to the thickness of the pipe.

 

[Philip Wood]

Delfador: Presumably this applies only for the pipe thicknesses much less than the dimensions of the magnet?

 

[Delfador]

Delfador: Presumably this applies only for the pipe thicknesses much less than the dimensions of the magnet?

Phillip Wood, I have only tested it for such thicknesses, yes. But I would assume it works for all thicknesses. The relationship I was talking about is v = m/T + c where v is velocity, T is the thickness, and m and c are constants, factors of the resistivity and the inner diameter of the pipe respectively.

This means that for an infinite thickness of pipe, the velocity will still be positive and equal to 'c'. This is based on my own observations. The equation doesn't take into account air resistance, which was negligible under the conditions of my experiment.

 

[gsal]

Sorry, guys, forgive my ignorance and arm yourselves with patience...I usually stick to the Software and Programming sub-forums, but with nothing new/interesting there for a couple of days, I ventured outside of it and found this rather curious topic...hope you don't mind me asking a few questions to understand it a bit further.

I will ask a bunch of questions in an attempt to corral my visualization of this phenomena; but, will formulate them so that most can be answered with a simple "yes" or "no"...expand if you wish.

Given a vertical pipe of inner diameter ID, varying diameter OD and a magnet:

given the vertical motion of the magnet, is it correct to say that only the horizontal component of the magnetic field comes into play, here?

Delfador's formula and message does not sound like arrived-to solution via analysis, is it?

Delfador says that m is a factor of resistivity and c of diameter.
does the mass of the magnet enter the equation?
does gravity enter the equation?

if you have a super conductive pipe:
can you drop the magnet into the pipe? or
will it be stopped in its tracks and kept levitated somewhere around the the pipe entrance?

if the magnet actually enters the superconductive pipe or can be made to magically appear already inside the superconductive pipe:
will it stay put at v=0?
will it drop at constant speed v=c as predicted by Delfador's formula? or
will it free fall at ever increasing velocity?

or put another way:
what exactly slows down the magnet? an opposing magnetic field? and
what holds the opposing magnetic field? a current? and
what holds the current? a resistive material?
what if the material has zero resistivity?

o.k., enough questions for a post...I have a few more for the case of a copper pipe; but let's stop here for now.

thanks.

 

[Philip Wood]

given the vertical motion of the magnet, is it correct to say that only the horizontal component of the magnetic field comes into play, here?

Yes, I think so. Its the field component perpendicular to the motion that matters.

Delfador's formula and message does not sound like arrived-to solution via analysis, is it?

Nor to me.

Delfador says that m is a factor of resistivity and c of diameter.
does the mass of the magnet enter the equation?
does gravity enter the equation?

Surely they must. Presumably D's formula purports to give the terminal velocity. At first the magnet must accelerate downwards until it is moving fast enough for the upthrust due to magnetic forces to balance the pull of gravity. The terminal velocity must depend on mg.

if you have a super conductive pipe:
can you drop the magnet into the pipe? or
will it be stopped in its tracks and kept levitated somewhere around the the pipe entrance?

The latter, I suspect.

what exactly slows down the magnet?

Some flux from the magnet's North pole passes out through the pipe walls and returns through the pipe walls to the South pole. Suggest you draw the set-up in section, including lines of flux. As the magnet falls currents are induced in the pipe walls in these two regions of cutting. Fleming's right hand rule shows that these currents follow the pipe walls round in circles. Because we now have current, AND a magnetic field at right angles to the current we have a Motor Effect (Lorentz magnetic) force on the pipe wall, and an equal and opposite (vertical) force on the magnet. The force opposes the magnet's motion, in accordance with Lenz's law.

The difficulty of analysing this mathematically to obtain a formula for the force, and thence the terminal velocity, is the curved paths of the flux lines and the non-simple way in which the flux changes as you go outwards through the pipe walls. If the walls are thin compared with the pipe diameter, the task would be a bit easier.

Ignoring any change in flux as we go through the pipe wall, and modelling the flux as confined, as it leaves the North pole, to pass through an axial length b of pipe wall, over which the flux density is constant [gross oversimplification!], and the same when it returns to the South pole, I find a terminal velocity of
$$v = \frac{mg\rho}{4\pi RB^2bT}$$
T is the pipe thickness, R is the pipe diameter, and ρ is the pipe material resistivity.
This predicts that the magnet won't move (v = 0) if the pipe thickness goes to infinity. Clearly this is rubbish because the outer layers of such a pipe would be too far from the magnet for its flux to pass through them. But we knew that this simple analysis won't work for thick pipes.

 

[gsal]

O.k., so, I went ahead and tried to think about the experiment. Please correct me if I am wrong.

Due to the opposing magnetic forces, I presume that if one drops even a spherical magnet into the pipe, the magnetic poles of the magnet will end up being vertical, i.e., one on top and one on bottom; any other orientation would yield differing forces that would tend to rotate the magnet, correct?

The other stable position that would produce equal forces all around, I think, is when the magnet is oriented exactly horizontal...but the position of equilibrium is unstable, correct? Meaning, a slight disturbance, and forces will probably rotate the magnet vertical.

So, I took a cylindrical magnet and positioned it vertically inside the pipe (north pole down) and tried to imagined how the magnetic lines go from north to south pole, how they cut the copper pipe and how they induce currents.

From what I can see, the currents are flowing in the horizontal direction on both sides of the cross section...in other words, the currents flow in a way that form horizontal rings that enclosed the inside of the pipe. The magnitude of these circular currents are higher at heights where the horizontal component of the magnetic lines are greater (at locations beyond the ends of the magnet) and become smaller towards the center of the magnet. See drawing.

is this interpretation correct, so far?

 

[Philip Wood]

You don't seem to have said "Thank you" for previous attempt to help.

 

[gsal]

Oh...sorry...I was caught up in the magnetic field ;-)

I do thank you for keeping up with me.

gsal

 

[Philip Wood]

Yes, I pretty much agree with you, as you can see from my long post (6). Here are thumbnails which should clarify the model I was using in that post.

 

[Delfador]

Hi guys, you're right when you say that my equation was not obtained by strict analysis. It is merely my way of explaining how the experimental results of the velocity of the magnet varies with the thickness of the pipe.

I hope you don't mind a few more questions.

The difficulty of analysing this mathematically to obtain a formula for the force, and thence the terminal velocity, is the curved paths of the flux lines and the non-simple way in which the flux changes as you go outwards through the pipe walls.

We know that the magnetic flux is the surface integral of the normal component of the magnetic field. If we know the dimensions of the magnet, can we determine the integral of the normal component of the magnetic field using a mathematical formula? Assuming that all dipolar magnets develop a similar pattern of magnet field lines.

Also, could not the same be done for the change of flux as you go outwards through the pipe? It would be a complicated task, for sure, but possible. Unless the behavior of the flux lines is different within the pipe and outside the pipe. But this shouldn't be the case because the magnetic permeability of copper and aluminium is close to that of vacuum.

If the walls are thin compared with the pipe diameter, the task would be a bit easier.

How thin do the walls have to be compared to the pipe diameter? The thickest pipe I experimented on had a thickness less than a third of the inner diameter. But to see the affect of the thickness of the pipe, we would need to have pipes of such thicknesses.

I am really interested in this phenomenon and I thank you guys for contributing to it.

 

[Philip Wood]

If we know the dimensions of the magnet, can we determine the integral of the normal component of the magnetic field using a mathematical formula? Assuming that all dipolar magnets develop a similar pattern of magnet field lines.

There is an equation for the magnitude of the flux density due to a magnetic dipole. You can find it by Googling 'Dipole'. It applies if you are much further from the dipole than the dimensions of the dipole. It would not, therefore, apply very accurately to the usual set-up of a bar magnet falling through a copper tube. Even so, it is quite a difficult equation to work with.

Also, could not the same be done for the change of flux as you go outwards through the pipe? It would be a complicated task, for sure, but possible. Unless the behavior of the flux lines is different within the pipe and outside the pipe. But this shouldn't be the case because the magnetic permeability of copper and aluminium is close to that of vacuum.

I agree, but I think you'd need to use numerical methods, rather than expecting to find an analytical solution.
May I recommend that – unless you've already done so – you see if you can use the grossly oversimplified model I put forward in my earlier posts (6 and 10) to arrive at the equation I give in post 6 – assuming it's correct. Doing this will give you a good feel for the Physics involved. Then you can consider refinements to the model.

How thin do the walls have to be compared to the pipe diameter? The thickest pipe I experimented on had a thickness less than a third of the inner diameter. But to see the affect of the thickness of the pipe, we would need to have pipes of such thicknesses.

For my oversimplified model to work reasonably well, I wouldn't want to make the pipe walls thicker than about a tenth of the tube diameter.