Off-axis magnetic field between two current-carrying rectangular loops

[ssj2poliwhirl]

Homework Statement

Not exactly a homework problem, but the actual working resembles it.
Basically it is a standard calculation for the magnetic field between a set of Helmholtz coils, except with a rectangular loop instead of circular ie: two identical rectangular magnetic loops (dimensions L x W) carrying a current I in the same direction, placed symmetrically along a common axis, one on each side of the experimental area, separated by a distance d.

I want to be able to calculate the total magnetic field due to the loops at ANY point in between them (off-axis magnetic field). I believe that for circular coils there is no analytical solution for this, but I *think* there should be for a rectangular loop.

Homework Equations

Applying Biot-Savart Law

The Attempt at a Solution

So at first we start with the same method for finding the on-axis field (which is relatively trivial) - find equations for the magnetic field caused by each loop separately and then sum them at a given point.

We consider each loop as 4 segments of straight, finite wire (2 pairs of length L and W with the appropriate orientation), noting that each pair's current is going to flow in the opposite direction.
Because of this fact, the on-axis field is easy because the magnetic field components off the axis for a line of wire cancels out with its opposite pair, so we just sum the 4 on-axis fields and we're done.

We can't do this for the off-axis calculation, so I think we need to start looking at perpendicular lengths and angles to calculate the field due to the wire at any point. I can do this in two dimensions (ie: considering the wire to be along the y-axis, and the point we are finding the field at in the x-y plane), but start getting a bit confused in three dimensions when it comes to adding up the components from each side of the rectangular loop - unlike previously, each segment's field will point in a different direction without cancelling neatly.

It occurs to me that the field's magnitude will be the same for a section of wire for the same perpendicular distance regardless of whether the point is neatly in the x-y plane or not, but I get a bit confused combining the different directions with other segments of wires.

Any advice or suggestions would be greatly appreciated!

 

[rude man]

I wouldn't be too sanguine about your prospects.

I do know that computing the self-inductance of a rectangular coil is very - er - problematic. And what is that but determining the total flux piercing the coil for a given current.

 

[ssj2poliwhirl]

Ah oh dear, really :(

I just thought that it would nicely break down into summing magnetic field components from 8 straight finite wires, which ultimately makes the problem finding the field at any point (x,y,z) due to a finite wire [I'm already not very comfortable with this though, since I've only ever seen it done in 2 dimensions] - with the difficulty coming from combining the three-dimensional field components, but I thought that was just my own lack of expertise with EM.

So was I oversimplifying the solution?

 

[rude man]

So was I oversimplifying the solution?

IMO yes, but there are wiser souls on this forum than I and I'm guessing that if they have any good ideas for you you will hear from them.

 

[paranormal]

I can understand your confusion and the challenges you are facing in calculating the off-axis magnetic field between two current-carrying rectangular loops. The Biot-Savart Law is a powerful tool for calculating magnetic fields, but it can become complex when dealing with multiple segments and orientations.

One possible approach to simplify the problem is to consider the rectangular loops as a combination of circular loops. This way, you can use the analytical solution for the magnetic field between two circular loops and then integrate it over the rectangular loop's dimensions to get the total magnetic field at any point. Another approach could be to use numerical methods, such as finite element analysis, to model the magnetic field and obtain a more accurate solution.

I would also suggest consulting with experts in the field or looking for research papers on similar setups to see how they have tackled this problem. Collaborating with others and learning from their approaches can often lead to a better understanding and solution.

Overall, I appreciate your effort and determination to solve this problem. Keep exploring and experimenting, and you will eventually find a solution. Best of luck!