# Force Between Two Plates

I have two plates and I want to charge them so they repel each other. To do this I ran opposing currents through the plates and to create magnetic fields which will cause the plates to repel.

Now, I need to find the force exerted on each plate. I am having a little trouble with this...

I considered the plates as parallel wires spaced very close together and used Ampere's Law to get the force of the magnetic field to be: B=u0*Js/2 where Js is the linear current density.

How do I get the force from that? I tried using Biot-Savart but I didn't get it to work out correctly. Any help would be greatly appreciated. Thanks

sophiecentaur
Gold Member
2020 Award
Could you state the problem more exactly, please?

Thanks for the reply. Here is the problem.

I have two parallel plates which a current is being ran through. This will create two magnetic fields around the plates. I run the current in opposite directions so the plates will repel each other. I want to find an expression to determine the magnetic force exerted on each plate as a function of distance apart and amperage through the plates.

I hope that clarifies my problem.

sophiecentaur
Gold Member
2020 Award
The formula for the force between two parallel wires, carrying currents, is given in this link.
http://theory.uwinnipeg.ca/physics/mag/node10.html

Would that be good enough? It would probably depend on how wide the plates were relative to their separation. You could perhaps tackle the problem by treating the plates as a series of parallel wires, to account for the finite width.

Thanks again. I did an analysis where I treated the two plates as two wires to get a rough estimate. I don't know how valid this is when compared to the two plates.

I also did the approach you recommended where I treat the plate as a series of parallel currents. This website describes the approach I used pretty well.

http://www.physics.uq.edu.au/people/jones/ph348/sols/sol3/node5.html" [Broken]

This seems like a logical approach. However, at the bottom of the page there is an expression for the magnetic force between the two plates. This doesn't have a term for the distance between the two plates; which I believe to be a vital component of the force. Maybe this is the force at the surface of the plate or something.

Also, the gap between the plates is about 10 microns and the length of the plate is about half a meter (width is about .4m). So the gap is very small compared to the length of the plate.

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The formula for the force between two parallel wires, carrying currents, is given in this link.
http://theory.uwinnipeg.ca/physics/mag/node10.html

Would that be good enough? It would probably depend on how wide the plates were relative to their separation. You could perhaps tackle the problem by treating the plates as a series of parallel wires, to account for the finite width.

Ah yes, this is the force per unit length between two infinite wires. Just as it doesn't make sense to calculate the total force between two infinite wires (it would be infinite), it also doesn't make sense to calculate the total force between two infinite plates. We usually consider infinite wires/plates because this simplifies the math and allows us to discard fringe effects. To get a good approximation for the force between two plates, it might be a good idea to consider them as infinite plates with a surface current $$\vec{K}$$. Then one could find the force per unit area. Finally, you just multiply by the actual area of the plates to get an approximation of the total force, neglecting the fringe effects.

Ah yes, this is the force per unit length between two infinite wires. Just as it doesn't make sense to calculate the total force between two infinite wires (it would be infinite), it also doesn't make sense to calculate the total force between two infinite plates. We usually consider infinite wires/plates because this simplifies the math and allows us to discard fringe effects. To get a good approximation for the force between two plates, it might be a good idea to consider them as infinite plates with a surface current $$\vec{K}$$. Then one could find the force per unit area. Finally, you just multiply by the actual area of the plates to get an approximation of the total force, neglecting the fringe effects.

Thanks for the help. This seems like a very good idea. How would I quantify the value of the of the surface current?

Thanks again. I did an analysis where I treated the two plates as two wires to get a rough estimate. I don't know how valid this is when compared to the two plates.

I also did the approach you recommended where I treat the plate as a series of parallel currents. This website describes the approach I used pretty well.

http://www.physics.uq.edu.au/people/jones/ph348/sols/sol3/node5.html" [Broken]

This seems like a logical approach. However, at the bottom of the page there is an expression for the magnetic force between the two plates. This doesn't have a term for the distance between the two plates; which I believe to be a vital component of the force. Maybe this is the force at the surface of the plate or something.

Also, the gap between the plates is about 10 microns and the length of the plate is about half a meter (width is about .4m). So the gap is very small compared to the length of the plate.

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sophiecentaur
Gold Member
2020 Award
I think this may be getting over complicated if all you want is an idea of the force.
If you do it with one wire, then with two (half current on each
) then with four etc. etc You'll reach a stage where the answers don't change much for the force produced. I think that's what you want.

The link at physics.uq.edu refers to an infinite pair of plates - the distance doesn't come into the formula because its the same whatever the separation. 'Our' plates (strips?) are of finite width. AS there won't actually be a precise solution the this particular problem, I suggest the pragmatic approach of gradually increasing the number of wires would at least get you somewhere near what is needed.

All this assumes that it's not just a theoretical question which someone has set and which requires the 'proper' answer. If this is the case them I'm not your man -I'm a practical engineer!

The two problems are (A) finding a model for the force; and (B) confirming the model. If it is possible to measure a change in pressure or deflection of the plates you have a means to evaluate your model. I am not sure how difficult a measurement would be ...

Thanks sophie. I was trying to find a way to expand on the idea that current flowing through two wires creates a magnetic field which cause the wires to repel or attact to plates. I don't know if this is possible, theoretically or practically.

I think what I will end up doing is wrapping the plates with a wire and running a current through the wire. This is a much more reasonable approach from the analytical side as I can just use the two wire method. If you think there will be any problems with this let me know. Thanks for the help.