Physical approximation to inverse square law using magnet(s)

[Swamp Thing]

What is the best way to create an approximately inverse-square-law magnetic field over a plane surface, e.g. complying with $1/r^2$ with less than 1% to 2% error over a plane annular region having R2 / R1 about 5 : 1 ? The goal is that a very small unmagnetized disk lying on the surface should experience a force compliant with this law to 1% over roughly that kind of region.

 

[Dale]

As far as I know this is not possible. A field that decays as $r^{-2}$ is a monopole field. There are no magnetic monopoles.

I think the lowest order would be a dipole-dipole force which should be $r^{-4}$ if I recall correctly

 

[renormalize]

What is the best way to create an approximately inverse-square-law magnetic field over a plane surface, e.g. complying with $1/r^2$ with less than 1% to 2% error over a plane annular region having R2 / R1 about 5 : 1 ? The goal is that a very small unmagnetized disk lying on the surface should experience a force compliant with this law to 1% over roughly that kind of region.

I'm having trouble visualizing your configuration as described. Can you supply a simple diagram?

 

[sophiecentaur]

a very small unmagnetized disk

etc. suspended on a thread could be more sensitive to deflection with no friction ?

As far as I know this is not possible. A field that decays as r−2 is a monopole field. There are no magnetic monopoles.

There is a 'school' demo, buried deep in my memory, which involved a magnetised rod, floating in a bucket of water with cork at one end. It orbited around a vertical wire with current flowing through it. The effect on the 'isolated' pole at the top was to make it follow the circular field round the conducting wire. I'm sure the details could be found somewhere with a good google search.

Using two long iron rods you could detect the repusion by the deflection angle of one rod, hanging down. 1/r₂ . could be measured over a limited range of separations. It would depend on how 'pure' the inverse square law had to be. You would certainly get a measureably different law for two short magnets - one suspended and one fixed. There would need to be a lot of experimentation to estimate the 'failure of complete isolation' due to the flux around the rods. The relative vertical placing would have a big effect, I guess.

If this is for a demo or a lab exercise, you'd need to find the best range of distances before presenting it.

 

[Meir Achuz]

A long magnetized rod looks like two magnetic poles, one at each end.
You can find the force law just as you would for two electric point charges. Just calculate how long the length of the rod has to be to have the distant pole contribute only one percent. 1% is fairly stringent, so it is going to have to be a long rod.

 

[Dale]

I wonder if there is a shape that you could put in a level tabletop that would make either the horizontal or tangential force follow a $1/r^2$ law. That might be easier than magnets, even if it is possible with magnets.

 

[sophiecentaur]

That might be easier than magnet

If it's not got to be magnets then I'm sure there's a mechanical arrangement. A 'suitable'* cam with a light thread passing over it would surely be one way.
*aye that's the rub

 

[A.T.]

I wonder if there is a shape that you could put in a level tabletop that would make either the horizontal or tangential force follow a $1/r^2$ law. That might be easier than magnets, even if it is possible with magnets.

You mean a potential well?
https://en.wikipedia.org/wiki/Gravitational_potential

 

[Dale]

You mean a potential well?
https://en.wikipedia.org/wiki/Gravitational_potential

Maybe. I am not sure that would give the right force as measured by an attached string, or if there would be a better way to measure the force

 

[A.T.]

Maybe. I am not sure that would give the right force as measured by an attached string, or if there would be a better way to measure the force

Since force is proportional to the gradient (slope) of the force potential, the force along the modeled potential well surface on an object under gravity should have the same relationship.

 

[sophiecentaur]

You mean a potential well?
https://en.wikipedia.org/wiki/Gravitational_potential

I did wonder about that but I though friction could be a problem.

Also, measuring the force would need to be done carefully to guarantee the 1% accuracy. I see that 1% is what a good 'test machine' could manage (a couple of hundred £, so that could make life difficult). Measuring the deflection of a pendulum could be cheap and accurate. But how to measure the actual point of contact?

There are many options but only for rough inverse square law confirmation, IMO.

 

[A.T.]

I did wonder about that but I though friction could be a problem.

You either build it as an air cushion table (like air hockey), or just use rolling balls.

 

[Vanadium 50]

Biot-Savart tells you that if you want an inverse square force, it needs to be between (IL) elements. That becomes a geometric problem since L needs to be small with respect to R, but large with respect to the other pole.