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.