How Do Electromagnets Repel Each Other in a Zero-Resistance Environment?

[Student149]

Consider 2 similar solenoids/electromagnets with appropriate iron core with the following parameters: core Length (L), core Radius (R), electromagnet wire diameter (w), number of turns of wire/winding layer on the core (L/w), number of layers of winding (K), total number of turns ((L*K)/w), final current (I), mass of each electromagnet (m).

Case 1:

Let the two electromagnets initially be held together end to end by some physical force inside a hollow glass tube horizontally, such that 2 north poles touch each other i.e. the distance b/w them is 0. Initially the current is 0. Now the current is switched on and when it reaches (I), the mechanical force holding together the electromagnets is released.

We are assuming the hollow tube they are held in offers 0 friction or resistance when they travel away from each other. Moreover, we are ignoring all other external factors like air resistance, and gravity etc.

Query 1: What is the formula to calculate much time (t0) would it take for the two electromagnets to reach a distance (d0) from each other when the mechanical force holding them together is released?

Query 2: What is the formula for force b/w the two magnets using the above parameters w.r.t the distance (d) or time (t)?Case 2: Instead of being horizontal the tube is now vertical and one of the electromagnets is pressing against the ground so only the top magnet can move. With the same initial setup, instead of distance we want to calculate the height (h) :

Query 1: What is the formula to calculate much time (t0) would it take for the top electromagnet to reach a height (h0) from bottom magnet's north pole, when the mechanical force holding them together is released?

Query 2: What is the formula for force b/w the two magnets using the above parameters w.r.t the distance (h) or time (t)?This is not some h/w Q but I am personally revisiting electromagnetism and looking for a blackbox formula. I have forgotten much of it so due apologies beforehand.

 

[berkeman]

This is not some h/w Q but I am personally revisiting electromagnetism and looking for a blackbox formula. I have forgotten much of it so due apologies beforehand.

What have you found so far in your searching and reading on these questions?

 

[Student149]

I think (though I am not sure) the magnetic field on one end is given by:

$${B_{\rm end} = \frac{\mu n I}{2\sqrt{L^2+(R+Kw)^2}}}$$

But I am not sure how this translates (and changes with distance/time) to force and the time (t0) required to travel a distance (d0) since the field must change with distance etc as both magnets move.

 

[Student149]

What have you found so far in your searching and reading on these questions?

any comments ??

 

[Steve4Physics]

I don’t believe there are any standard (‘black box’) formulae you can use. A quick Google search supports this. A mathematical analysis looks extremely difficult – possibly infeasible. (I’m sure someone will correct me if I am wrong though!)

Also, note that your formula in Post #3 gives only the magnitude of the field at an end. But the field from an electromagnet extends outwards in 3D, changing in magnitude and direction. It needs to be represented by a vector field. That means each part of the other electromagnet will experience a field different to the field at an end. That’s what makes the maths so hard.

 

[Student149]

I don’t believe there are any standard (‘black box’) formulae you can use. A quick Google search supports this. A mathematical analysis looks extremely difficult – possibly infeasible. (I’m sure someone will correct me if I am wrong though!)

Also, note that your formula in Post #3 gives only the magnitude of the field at an end. But the field from an electromagnet extends outwards in 3D, changing in magnitude and direction. It needs to be represented by a vector field. That means each part of the other electromagnet will experience a field different to the field at an end. That’s what makes the maths so hard.

Thanks for the reply. I too couldn't find much on what seems like an obvious/standard problem, hence the Q :(. I am fine with approximations with some margin of error (if we know any). I do understand the 3D part problem but I was hoping we have some idea of rough approximations (for a standard shape like a cylinder)..

 

[Steve4Physics]

Thanks for the reply. I too couldn't find much on what seems like an obvious/standard problem, hence the Q :(. I am fine with approximations with some margin of error (if we know any). I do understand the 3D part problem but I was hoping we have some idea of rough approximations (for a standard shape like a cylinder)..

If this is for an experiment with existing coils, I would adopt an empirical/numerical approach.
Measure the repulsive force as a function of separation for a given current.
Then, using the results, calculate accelerations, velocities and finally a set of distance-time data.
Repeat for a range of currents.

Alternatively, to get the force-separation relationship, each electromagnet (A and B) could be considered as a set of simple (circular) current loops, each of a certain radius and position. If you had the actual dimensions and a formula for the force between two current loops (not necessarily of the same radius), then a relatively simple piece of software could sum the force contributions of each of A’s loops on each of B’s loops.

Apart from that I’ve got no other suggestions.