Can I Use Circular Loop Formula for Magnetic Field of a Square Loop?

[KoGs]

I have a magnetic dipole moment of a known magnitude and direction (which I calculated). I need to find out what the magnetic field is at a point p away from my loop. This point is exactly perpendicular my loop.

I have a formula here that says B(p) = (permeability constant * magnetic dipole moment) / (2pi * p^3).

However this formula was derived by assuming my coil is a circular loop. My loop is a square. Can I still use the same formula? Thx for any help.

 

[OlderDan]

I have a magnetic dipole moment of a known magnitude and direction (which I calculated). I need to find out what the magnetic field is at a point p away from my loop. This point is exactly perpendicular my loop.

I have a formula here that says B(p) = (permeability constant * magnetic dipole moment) / (2pi * p^3).

However this formula was derived by assuming my coil is a circular loop. My loop is a square. Can I still use the same formula? Thx for any help.

For all points far away from the loop, the dipole field is a good approximation to the actual field. The actual field can be expanded in a series that would be called a multipole expansion. The dipole approximation discounts the higher order terms in the expansion, keeping only the dipole term.

 

[KoGs]

I'm confused what you are saying.

Are you saying my dipole moment is actually equal to my magnetic field?


And what about inside the loop? Is the magnetic field the same throughout?

 

[OlderDan]

I'm confused what you are saying.

Are you saying my dipole moment is actually equal to my magnetic field?


And what about inside the loop? Is the magnetic field the same throughout?

I am not saying the dipole monent is the field from the dipole. I am saying that the equation that gives you the magnetic field produced by a dipole at large distances away from the loop is a good approximation to the actual magnetic field regardless of the shape of the loop. The field near the loop, and at points within the loop will not be well represented by this equation. For the field nearby and inside, you need to use the Biot-Savart Law to calculate the field by integrting around the loop.

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/biosav.html

The field is not constant within the loop. These calculations are generally not easy except at points of symmetry, such as along the axis of a circular loop. Even at the center of square loop it takes some work to get the result

Is this the problem you are trying to solve?

https://www.physicsforums.com/showthread.php?t=120083

 

[KoGs]

That's not quite the question, but I think we can work with that. My loop is a complete square of known length and width. My p is in the loop, but not directly in the middle. The location of p inside the loop is also known.

Where in the Biot-Savart law does it factor in where p is inside the loop? You would have to integrate over the entire length of the wire right? But P isn't directly at one end of the wire.

 

[KoGs]

This is what I did: http://members.shaw.ca/KingofGods/physics.jpg
Answer in the back is -0.20 mT.

Oops I noticed I forget to put in the exp 2 in the first part. But I did put it in in the calculations.

 

[OlderDan]

This is what I did: http://members.shaw.ca/KingofGods/physics.jpg
Answer in the back is -0.20 mT.

Oops I noticed I forget to put in the exp 2 in the first part. But I did put it in in the calculations.

It looks like you are on the right track. I haven't absorbed all of your work, but it looks to me like your intergration limits are not correct. For the top wire, your definition of s leads me to believe you are integrating from s = -.02m to s = +.06m, not from 0 to +.08m. That means that on the average you are closer to the source than you assumed, so your contribution from the top wire will be greater. I'm not sure what you did with the other wires, but my first impression is that you have a symmetry that makes it necessary to only compute the contribution from 2 wires and I think you used that. It also appears that you are subtracting one contribution from another. That would not be right. All four wires contribute in the same direction.

 

[KoGs]

I'm confused what you mean by integrating from s = -.02m to s = +.06m. Oh, are we to assume P is on the 0,0 point?

I subtracted 2 wires from the other 2 because the i's are in different directions, so the B's would be in different directions as well?

 

[OlderDan]

I'm confused what you mean by integrating from s = -.02m to s = +.06m. Oh, are we to assume P is on the 0,0 point?

I subtracted 2 wires from the other 2 because the i's are in different directions, so the B's would be in different directions as well?

I looked at P as you had it in your diagram, 2cm from each of two wires along a diagonal of the square. The s that you drew is zero at a point 2cm from one end of the wire, so the wire ranges from s = -2cm to +6cm.

Think again about the field directions. Apply the right hand rule to each of the four wires.

 

[KoGs]

If that's the case then there is no symmetry. Both the top wire and the right wire would be integrated from -0.02 to 0.06. But R for the top wire is 0.02, and R for the right wire would be 0.06. And for the left wire, it would be integrated from -0.06 to 0.02, with R = 0.02.

 

[KoGs]

Nevermind. I worked it out. There is symmetry between the top wire and the left wife. But I still don't get the correct answer, even when I take the point P to be 0,0.

 

[KoGs]

Here is the work after making the corrections you suggested. http://members.shaw.ca/KingofGods/physics2.jpg

In fact subtracting the 2 gives me a closer answer to the one they have in the back of the book lol. So I must be way off here. Any further help would be much appreciated.

 

[OlderDan]

Here is the work after making the corrections you suggested. http://members.shaw.ca/KingofGods/physics2.jpg

In fact subtracting the 2 gives me a closer answer to the one they have in the back of the book lol. So I must be way off here. Any further help would be much appreciated.

I got their answer except for the sign, but that is just because I took B positive into the page, which is probably the negative z direction. I do not understand how you get your factor out in front. Why do you have 2pi instead of 4pi in the denominator? The very first time you wrote it you had 4pi and then it became 2pi. The rest of it looks correct, but I did not check all your numerical work. The factor of 2 would make your summed net field reduce to .22mT. I did the same things you did and got .1997mT. I think you are wrong about the 2pi and just need to check your numerical calculations again. The net field should definitely be the sum of the terms.

 

[KoGs]

Haha, you're right. Thx for the help.

What would happen if say we weren't inside a loop. But we still wanted to calculate the magnetic field generated by a wire that isn't infinite. These calculations we used here assumed the point we are trying to calculate at is perpendiuclar to the wire.

What if say the wire ran from 0,0 to 0,5. And our point we are looking at is at -2,7.

 

[OlderDan]

Haha, you're right. Thx for the help.

What would happen if say we weren't inside a loop. But we still wanted to calculate the magnetic field generated by a wire that isn't infinite. These calculations we used here assumed the point we are trying to calculate at is perpendiuclar to the wire.

What if say the wire ran from 0,0 to 0,5. And our point we are looking at is at -2,7.

The approach would still be the same. There is a plane that contains the wire that also contans point P. The field will be perpendicuar to this plane. There is a perpendicular distance, R, from the point where you want the field to the line that contains the wire. It is just a matter of finding the correct limits of integration. Of course that is for a single wire. If you have multiple wires, each wire might have its own plane tha contains P. If you wanted to find the field at a point that was not in the plane of your loop, you would have different directions for the different segemts of the loop and would have to add the vector field contributions from each of them.

 

[KoGs]

So in the example I just gave the limits of integration would be... 2 to 7? Or would it be from -7 to - 2 ?

 

[OlderDan]

So in the example I just gave the limits of integration would be... 2 to 7? Or would it be from -7 to - 2 ?

-7 to -2 with R = 2

 

[KoGs]

Could you explain why one and not the other? Thanks.

 

[OlderDan]

Could you explain why one and not the other? Thanks.

Limits from 2 to 7 would give you the same result by symmetry (wires represented by ==========)

-------------------------------------P--------------------------------------

================ . . . . . . . . . . . . .================

either wire gives you the same field