# Closed loop shapes with zero mutual inductance?

I have done numerous searches trying to find any mention of theory or experimentation done on determining closed current loop shapes that have zero mutual inductance and...no dice!

Of course you can do things like put two loops infinitely far apart or surround both in thick mu-metal, closed shells, but things like that are not what I'm talking about.

What I am talking about is taking two or more finite length conductors, each with a head and a tale, and shaping them such that:

• They don't intersect themselves before head and tail meet
• They don't intersect other conductors
• There is not any point where a conductor passes infinitesimally close to itself or another conductor without actually intersecting
• And any other rules that allow each conducting loop to have its own self inductance without interacting with the other loops.

This is the equivalent of zeroing the Neumann mutual inductance equation: $$M_{m,n} = \frac{\mu_0}{4\pi} \oint_{C_m}\oint_{C_n} \frac{\mathbf{dx}_m\cdot\mathbf{dx}_n}{|\mathbf{x}_m - \mathbf{x}_n|} = 0$$ But still allowing self inductance to be greater than zero:

$$L_{m,m} = \frac{\mu_0}{4\pi} \oint_{C_m}\oint_{C_m} \frac{\mathbf{dx}_m\cdot\mathbf{dx}_m}{|\mathbf{x}_m - \mathbf{x}_m|} > 0$$

If any of you have insight that could help with this curiosity, I'd love to hear it.