# Magnetic flux linked between two coils placed co-axially

I came across a scenario in which a coil having a radius $$r$$ is placed co axially to another coil having radius $$R$$. They have some non zero finite distance between their centers. If some finite current is passed through the coil of radius $$r$$,(provided that $$r<) what is the magnetic flux linked with the larger coil due to the magnetic field produced by the smaller coil.

I tried to equate the flux density to solid angle but realized that the dipole did not produce magnetic field which was symmetrical in all directions, i.e. the flux density was not constant in every direction at a given constant distance from the dipole.

I also tried to integrate the magnetic flux by taking the product of magnetic field at every point in the plane of the larger coil, but found it too complex to compute.

Is their any way to form a mathematical model of this situation which is easy to compute as well as to visualize? Your opinions are welcome...

• This is not a trivial problem even with your approximation as can be gauged from this old but still useful paper. – Farcher Sep 3 at 6:16

G. Smith gave the answer (brute force method) but there is a trick to do the problem other way around which is usually done on UG level. Pass current in bigger coil and find the mutual inductance ($$M$$) between the two coil. The flux linked due to current in smaller coil will simply the product of $$M$$ and current in smaller coil. I would give one simple calculation of $$M$$ $$\phi_{\mathrm{link}}=\int\vec{B}•d\vec{a}$$ $$\approx\vec{B}•\int d\vec{a}$$ Magnetic field stays constant over the whole area of smaller ring since it's area is very small $$\frac{\mu_{0}}{2}\frac{I R^2}{(z^2+R^2)^{3/2}}\pi r^2$$This is equal to $$M$$ times current in larger coil you can now proceed further.
Refinement: if you want to decrease the distance between the two coils so our area approximation doesn't hold anymore then you can simply go for multiple expansion. And if you're champion of integration Neumann formuale is there for you $$M=\frac{\mu_0}{4\pi}\oint \frac{d\vec{l}_1•d\vec{l}_2}{\mathcal{r}}$$
If $$r \ll R$$, you can model the smaller coil as a point magnetic dipole with magnetic dipole moment $$IA$$. ($$I$$ is the current in the loop and $$A$$ is the area of the loop.) The field of such a dipole is reasonably simple.