# Need help doing force calculations with magnetism (between magnets)

I am trying to calculate the forces (in Newtons) between a permanent magnet, and an electromagnet. All I can find is interactions between permanent magnets, electromagnets, but never both and it is getting really confusing for me to do the calculations.

So basically, if I have a permanent magnet of strength X (in Teslas), and an electromagnet of strength Y (again, in Teslas) and a distance Z (in meters), what will the force be between them? If more variables are needed, let me know.

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"How much force" depends upon each magnet's field strength, where they are in relation to one another, the magnetic permeability of what ever is in between them, and their "geometry" (physical shape, dimension and construction). The math is easy enough when considering ideal magnetic point charges, but this last part is where most of the complexity resides, and renders easy answers to real world situations impossible.

Very well, I will try to simplify it further. Let us say that the magnetic field strength is B in Teslas. Also the size of the filed is sufficiently large such that it appears to be linear locally, therefore the effects of curvature are unnoticeable. So if we were to place a magnetic coil where the field is strength B, with a radius of r, current of A, and N number of loops, what will the force F be? And don't worry about angles, it will be aligned to provide the maximum force given these conditions.

I am trying to calculate the forces (in Newtons) between a permanent magnet, and an electromagnet.
This is valid only for the case when both magnets have the same area, and have a small air gap between them, but,

where B= flux density in Teslas, A = area (square meters), and μ0 is permeability (4π×10−7 T·m/A in free space)

The general form for point interaction is:

where qm1 and qm2 are magnetic pole moments in ampere-meters, r is separation distance in meters, and μ is the permeability.

You already know the flux density of both the electromagnet and permanent magnets. Nearly everything else depends on the geometry between them.

How about building a spring scale test rig, and measuring it?

jim hardy
No no no, that will not do. I need something where I have an electromagnet with a radius of r, coil windings of N, and current of I. Now the size of the permanent magnet is huge, say, around the size of a planet. Let us use the magnetic field of the earth for example. So since we know the approximate value of the magnetic field strength of the Earth in Teslas, what would the force be given these conditions?

And I wish I could test something like this but I don't have the money.

Would it be fair to say you are looking for an equation you can plug the above values into that yields an answer for the amount of force?
AFAIK, no such equation exists.

marcusl
Gold Member
Since the planetary field is uniform over the size of your electromagnet, there is indeed an equation for this situation: F = 0. There is no net force on a loop of wire, or an assembly of loops, in a uniform field.

There is a torque that will tend to align the coil axis with the incident field.

Asymptotic and jim hardy
What do you mean when you say uniform field? And would an electrical coil create non- uniformities? You can't tell me there is no force between electromagnets and ordinary magnets. If so electrical generators would not work.

I know there would be a torque, but what if it is pole facing pole?

marcusl
Gold Member
What do you mean when you say uniform field?
That the direction and strength of the H field is constant at all points across your electromagnet. That is what you have when you specify using the earth's magnetic field.
You can't tell me there is no force between electromagnets and ordinary magnets.
I won't try to tell you anything you don't want to hear.

jim hardy
I won't try to tell you anything you don't want to hear.
Well sorry if my frustration is coming through in this. However, science does not care about how either of us FEEL about things. But that is alright. The sooner we know what is wrong the sooner we know what is right and start adapting.

But I think my problem is in how I phrased my question. So I will split it into two parts.

1) If you were to stand at either the north or south pole, and you have a magnet with you and you and you face it having the same pole facing the geological one of the earth (north/north or south/south) are you saying no upward (away from the earth) force will be created? And if it can be done, why can't it be done with an electromagnet?

2) Let us say we have two bar magnets and one is attached to the other by means of a lever that pivots around the center of one of them. One of the magnets were free to rotate while the other has one of the poles facing the magnet and the other is facing away. Now let us say the bar magnet that the lever is hinged to is not allowed to rotate because it is held stationary, or is much larger and more massive, or both. This leaves only the lever and the smaller magnet free to rotate. Now keep in mind that the lever can freely rotate around the more massive magnet, but not the smaller one relative to the lever therefore both the lever and smaller magnet must rotate as a single component. Now if this lever were rotated to the side of the larger magnet resulting in a 90 degree angle between the poles of the smaller magnet and the larger one, what would happen when it is released? How will it rotate?

....

2) Let us say we have two bar magnets and one is attached to the other by means of a lever that pivots around the center of one of them. ....
I got lost. Diagram please (labeled).

I got lost. Diagram please (labeled).
I'll load a diagram later, but perhaps I can have another way of explaining it.

Say we have a quadcoptor drone hovering above the magnetic equator of the earth with a sufficiently powerful electromagnet mounted to it. Now keep in mind that the poles of the electromagnet is pointed straight up and down. However when the electromagnet activates, it runs on DC power and it creates a torque. However, when a torque is created, the quadcoptor adjusts its engines to counteract the torque exerted while still maintaining altitude. So will it just hover there or will it move to the north or south based upon polarity?

jim hardy
Gold Member
2019 Award
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So basically, if I have a permanent magnet of strength X (in Teslas), and an electromagnet of strength Y (again, in Teslas) and a distance Z (in meters), what will the force be between them?
I think zero in direction of fields, but there is torque as has been said earlier.

A permanent magnet is a dipole so one end of it pushes and the other pulls against earth's field netting zero.
But it will try to align like a compass because of the torque..

https://van.physics.illinois.edu/qa/listing.php?id=1027

https://van.physics.illinois.edu/qa/listing.php?id=1027
Q & A: Magnets floating in the Earth's Field

Q:
People say that the earth is a big magnet. We are wondering if a strong piece of magnet can float in the air, based on the principle that like poles of a magnet repels each other.
- Wu Fan and Qihan
A:
Hi Wu Fan and Qihan,

That is a very good question, because it addresses a feature of magnetic forces which normally does not arise in our experience with small magnets.

The force on an object is related to the change in the energy of a system (not including the kinetic or thermal energy of the object) when the object is moved. We write

F = (change in Energy)/(change in position)

For static fields. The change in position has a direction, and so the force does too (you need some vector algebra with a dot product to express this exactly).

Two small magnets placed together with like poles close to each other feel a repulsive force because of the energy stored in the magnetic field. The energy density in space is proportional to the magnetic field squared, and when the close-by poles are the same, their fields add in more places than they subtract, and so the total energy is higher for this case than when opposite poles are closer, where the field is smaller in more places.

There are two things about the Earth’s magnetic field which makes this effect much smaller. For one, the field is very weak at the surface (about a gauss or less). The more important reason is that because the field extends over such a large space and because we on the surface are far away from the center of the Earth’s dipole, the Earth’s magnetic field strength is very uniform if you look at it over a region of space that is reasonable in size (like the size of the magnet you propose to use).

If you put these two pieces together, you find that the force on a magnet due to the Earth’s field is very small -- if you move the magnet from one place to another, its field adds to the Earth’s field in almost the same way because the Earth’s field is very little different from one place to another, and the total magnetic energy changes by a very very tiny amount. In fact, the total magnetic force on a magnet in a uniform magnetic field is exactly zero, and the forces we normally associate with magnets repelling or attracting are proportional to the rate of change of the field strength with position.

This isn’t the end of the story, however, because the magnetic energy of the system depends on which way the magnet is pointing, relative to the Earth’s field. If it points along the field, the fields add, for a higher energy. If it points the other way, the fields subtract, for a lower energy, and so the magnet prefers to turn to point in this way. Magnets in uniform fields feel torques which make them turn around if they are not pointing in the right direction, but there is no net force making the magnet want to levitate.

That having been said, if you had a really really big magnet, whose field extended over such a large region that the Earth’s field changes noticeably over that region (you might need another Earth-sized bar magnet), then yes, a noticeable force can be produced.

As for actual levitation, that can only happen with materials whose magnetic moment actually points the wrong way, increasing the energy in a magnetic field. These are called diamagnets. Diamagnetism is purely a quantum mechanical effect, with no classical explanation. By far the most intense diamagnets are superconductors. You may have seen superconductors levitating over magnets, or vice-versa. The Earth’s magnetic field does not change rapidly enough from place to place to levitate even a superconductor.

Tom (w Mike)

berkeman and Asymptotic
I think zero in direction of fields, but there is torque as has been said earlier.
So wait, you are telling me that two magnets will with their poles facing one another will not be attracted or repulsed? Really? You mean all those elementary school experiments with magnets are just my imagination?

jim hardy
Gold Member
2019 Award
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So wait, you are telling me that two magnets will with their poles facing one another will not be attracted or repulsed? Really? You mean all those elementary school experiments with magnets are just my imagination?
oops, of course not

i was back to earth's field. If your permanent magnet is located at earth's center and your electromagnet at earth's surface
you can't make much force, see Asymptotic's equation in post #4
where qm1 and qm2 are magnetic pole moments in ampere-meters, r is separation distance in meters, and μ is the permeability.
Calculate force on each end of your electromagnet. What is difference between r's for its two ends?
If it's a meter tall,
r1 = 6.371 X 106 meters
r2 = 6.371 X 106 + 1 meters

here's what i get for the ratio of the two forces

Can you feel two magnets even at arm's length apart ?

Best i ever did was build a magnetometer that from my kitchen table could sense a car in my driveway about twenty feet away.

old jim

marcusl
Gold Member
CCatalyst, what is your education in physics, particularly electromagnetism? Do you know vector calculus? The Lorentz force? Your questions imply an approach that is not well grounded in physics principles and math.

CCatalyst, what is your education in physics, particularly electromagnetism? Do you know vector calculus? The Lorentz force? Your questions imply an approach that is not well grounded in physics principles and math.
Really? I was educated in aerospace engineering. I WAS asking for the arrangement of variables.

Anyway I've slowly learned over the course of this thread that magnetic repulsion/attraction rely upon non-linear magnetic field lines. And yes, I am aware of the Lorentz force. (current_x*magnetic field_y=force_z) But I didn't think I needed to focus on it until now.

So here is a new scenario. We take a completely uniform magnetic field until we insert a superconducting plate at a normal angle relative to the field allowing a maximum magnetic field deflection for its shape thanks to the Meissner effect. Once that is done, we place a solenoid in front of the superconducting plate where the magnetic field is no longer linear because of the Meissner effect. Now, if we were to send a direct current through the solenoid, what would happen? Would there be a force exerted on the magnetic field generator? Would it be exerted upon the superconducting plate? Would it be on both? Let me know.

marcusl
Gold Member
I asked because your posts contained no equations, no math and no apparent understanding of physics, and referenced elementary school experience. If you have some background, then put it to use instead of concocting ever more byzantine scenarios with levers and whatnot. The vector form of the Lorentz force is $\mathbf F=q\mathbf v \times \mathbf B$ (don't write just one component, and use symbols instead of words). If you apply it to a current loop whose plane is normal to that of the uniform field, you should be able to show yourself that F=0 when integrated around the loop (think of the right-hand rule if you can't do the integral). The same is true for each of a stack of loops, naturally. The currents interconnecting the loops to form a solenoidal coil lie parallel to the field, so the Lorentz force equation tells you what force they contribute (hint: zero). Now reason for yourself what happens if the electromagnet coil is in a nonuniform field coming from a localized source. Then apply some physics reasoning to the half-dozen other scenarios you've made up.

NTL2009, Asymptotic and jim hardy
I asked because your posts contained no equations, no math and no apparent understanding of physics, and referenced elementary school experience.
First of all, this is not the place to judge people based upon assumptions and speculation. Second, how am I supposed to know what the equation is before asking what it is? How is that even possible? You honestly think I did not try to look this up first? And I was starting to get the feeling this may be a vector calculus application, and if so just say so. Plus I already knew about the Lorentz force long before this thread. That calculation is EASY! Third, I was not implying that I have an only elementary school level experience of magnets. I was pointing out how it almost sounded like people were saying that larger magnets (next to the one of the same size) were actually weaker. And that is what I felt contradicted even elementary school level knowledge on the subject. Now I know, thanks to Jim Hardy here, that is is actually a bit more complicated than that. He actually explained things to me and actually succeeded in advancing my knowledge of the subject. Fourth and finally, just because I have a background in aerospace doesn't mean I know everything, and there is no arrogance in admitting that, quite the opposite in fact. Presently there is not much overlap between aerospace and electrical engineering. But I think I may have hit on something that would need that to change someday and that is why I made this topic.

So could you please stop treating me like I'm faking knowledge for attempting to further my own? It would be appreciated.

Now getting back on topic, I think I may have figured something else out. I'll make a diagram of this soon. Let's start with a plate made of a superconducting material with a coil in front and behind that carry counter-rotating currents. The superconducting plate is placed at a 90 degree angle to a magnetic field, causing great curvature of the field much like a plate deflecting an aerodynamic flow. Now, is it possible to place the coils in such a way that would result in a net Lorentz force parallel to the magnetic field lines if they were undisturbed? And more importantly, as per Newton's third law, what would this force push against? The magnetic field source? The plate? Both?

marcusl
Gold Member
Ok, sorry. We sometimes have troubles with posters who ask a lot of questions just to hear themselves talk.
Now getting back on topic, I think I may have figured something else out. I'll make a diagram of this soon. Let's start with a plate made of a superconducting material with a coil in front and behind that carry counter-rotating currents. The superconducting plate is placed at a 90 degree angle to a magnetic field, causing great curvature of the field much like a plate deflecting an aerodynamic flow. Now, is it possible to place the coils in such a way that would result in a net Lorentz force parallel to the magnetic field lines if they were undisturbed? And more importantly, as per Newton's third law, what would this force push against? The magnetic field source? The plate? Both?
You can analyze this. Assume two identical single-turn loops (because they are easiest to analyze and because it's easy to find field plots online) that are coaxial. Since interactions fall off quickly with separation, place them fairly close together--separate them by, say, one diameter. Omit the superconductor to start with. Now, look at the B field from loop 1 here
https://online.science.psu.edu/site...electron/500px-VFPt_dipole_magnetic3.svg_.png
or in the plot of your choice. Draw coil 2 one diameter to the right of coil 1, and notice how the field lines curve at the wire of loop 2. Keeping in mind the opposite current flows, apply the Lorentz force rule and tell us whether the force is attractive or repellent.

To take step towards including a superconducting plate, think about what the field is in the plane parallel to the loops and halfway between them (you'll get the answer from symmetry and from the opposite signs of the currents).

To take step towards including a superconducting plate, think about what the field is in the plane parallel to the loops and halfway between them (you'll get the answer from symmetry and from the opposite signs of the currents).
Well I keep figuring that both coils will want to go the same direction. The question I have is if they were attached to a frame that also had the superconducting material attached to it, would the frame want to move in a relatively uniform direction or would the assembly just try to pull itself apart? Or could it be a little of both? Again, I'll draw a diagram when I have the time.

Edit: Also keep in mind there is an additional magnetic field going across the entire device from a big magnet from far away, and the field would appear to be virtually linear had the apparatus not been placed in it.

marcusl
Gold Member
So you still have not attempted to solve the problem by applying physics principles, and seem uninterested in learning what they are. I am shifting my attention to other posters who are.

Last edited:
berkeman
Mentor
@CCatalyst -- I think the issue with the uniform magnetic field interacting with a small magnet is that it attracts one pole and repels the other pole about the same, so there is no net translational force. I'm not able to tell if you are understanding that point yet. I hadn't really thought about it until reading through this thread.

Asymptotic and jim hardy
So you still have not attempted to solve the problem by applying physics principles, and seem uninterested in learning what they are. I am shifting my attention to other posters who are.
That is simply not true. I do realize now that no net magnetic force is possible in a completely uniform magnetic field. And I slowly figured that out thanks to THIS topic and my physics textbook.

But I was not asking that in my later posts. What I was asking is if I could generate force using by introducing non-linearities or curves in what would otherwise would be a uniform field. I guess I'll have to provide a quick sketch to visualize what I am describing. It may be crude, but it will do.
F:\Pictures\Lorentz_Meissner[/IMG]
If you can see this image basically the field lines START as being uniform but then are forced to curve thanks to that shaded block of superconducting material in the way. As it travels around while being forced into angles as far from uniform as possible, they travel through coils that have their currents represented by x's and dots showing they are rising and entering the paper. Now all I want to know is if this setup will result in any net force and on what components. And for the last time I know now that a uniform magnetic field cannot create any force. I get that.

Edit: By the way, is there any way for me to upload images from my computer to this forum?

...Edit: By the way, is there any way for me to upload images from my computer to this forum?