I understand that the definition of flux linkage as kind of the sum of all the fluxes going through the surfaces bound by all the turns. That is, $$\Lambda = N \Phi$$. However what I don't understand is what happens if the coils are not uniformly stacked up on top of each other like a solenoid. For example, take the example of a sphere with N turns of wire wrapped around it to create some sort of spherical solenoid and run a time varying magnetic field through it. In this case, there should be a voltage induced however I don't understand how to employ the concept of flux linkage to calculate that here because $$\Lambda \neq N \Phi$$ as every loop of wire links a different amount of flux to the current. If anybody could help me understand how to use linkage here to solve for induced voltage or self inductance that would be great.
• $N \Phi$ is a simplification of flux linkage to the case where all the wire loops (fictitious or otherwise) have the same shape and the same magnetic field penetrating that shape. If you don't have that case, you have to sum the flux linkage from each loop or integrate depending on the limit you are faced with. – Finncent Price Aug 8 at 8:36
• I think you want to compute the area of each coil using something like: A = $\pi R^2 sin^2 \theta$ for each loop and then integrate that over the whole sphere. I assume the magnitude of B is the same for all the loops. – Finncent Price Aug 10 at 11:52