Hello, my problem here is kind of special and I have no clue, if I did the physics right (probably not).

I want to find the magnetic field of a wire with the "boundary equation": **(a\*cos(t-δ\*sin(t)),a\*h\*sin(t))**. This shape can be seen in Picture 1 and 2. As if this alone wouldn't be hard enough, there is more current flowing on the left side of the wire, then on the right side.

(The wire here is represents a plasma in a tokamak fusion reactor, but this doesn’t really matter)

I tackled this problem in two steps: 1) find the direction of the magnetic field vector B at position r,t (r = radius from the center; t = angle from 0 to 2pi) 2) find the magnitude of B

1) As you can see in Picture 3 I assumed, that the direction of B can be calculated, by an vector r, that at an radius/distance from the center r (sorry for my notation) does a small step in angle t. (it runs along the curve) Then we calculate the vector at angle t to the vector at angle t+dt. -> **B(r, t) = r(t + dt) - r(t)**. Although this approach seems very intuitive for me, I haven’t found a formula which describes this.

2) (See picture 4) The first thing we do here is normalizing the Vector B. (Bp stands for B prescaled and should just note, that it is not the final vector B ) Then we have to find the actual magnitude of B. The current density in the wire is described by 2 parts. The homogeneous current density J0, like in a normal wire plus the "shift in current". I have honestly no idea, what the shift in current term should really look like. In my "calculations" I just declared it as **u \* J0 \* -cos(t) \* r**, because it looks intuitive ( -> that means it is probably wrong). "u" is just a factor between 0 and 1, which scales J0. -cos(t) is negative at angles of the left side (0°) and positive at angles of about pi, or 180° at the right side. So this is the key point for the shift. (in my "calculations" I forgot the minus sign). The factor of the radius "r" should just smoothly transition between the left and right side. If we then solve for B (based on the formula of picture 5) we get the equation at the bottom of picture 4. I know pi\*a² is a really bad estimate for the Area of this weirdly shaped ellipse, but I don’t know, how to actually calculate the area, given this curve: **(a\*cos(t-δ\*sin(t)),a\*h\*sin(t))**.

  
outside the wire:

B outside seems to be B inside \* (a² / r²) (a = total "radius" of the boundary of the wire, like in equation from top, r = radius from the center of the midpoint). If we apply this very stupid observation, we get the B, shown in picture 6.

 

**Pretty much all of these calculations here are made up, without any usage of known formulas, so there are probably completely wrong. I would appreciate it, if someone, who unlike me really understands these physics could give me a feedback on my approach an correct my calculations. Thank you.**

To be honest: "My" idea for the "shift in current" term with the cos(t) factor, is from an LLM, which had to bring my thoughts in the right direction.

 

The magnetic field inside an tokamak reaktor plasma