Magnetic flux with two concentric cirular loops

Question

Suppose I have two cocentric circular loops, one with radius $R_2$ and one with radius $R_1$ where $R_2$ is the radius of the larger loop. Current is flowing in the outerloop and is changing over time, with some value of $dI/dt$. The moving charges generate a magnetic field $B(s)$, the magnitude of which is given by $$B(s)=frac{mu_0I}{2s}$$
where $s$ is the distance from the outer loop. The magnetic flux $Phi$ through the inner loop is given by $$Phi = intfrac{mu_0I}{2s}cdot da$$
and I've chosen to do this in polar coordinates. Then $da = sdtheta ds$, and I have for the flux $$Phi =frac{mu_0I}{2}int dtheta ds$$
where $theta$ ranges from $0$ to $2pi$ and $ds$ ranges from $R_2-R_1$ to $R_2$. But this would just give the area of the inner loop. This is not correct, as it does not reflect the fact that magnetic field is not constant in the inner loop. How can I fix this?

Young Kindaichi · Answer

The expression
$$B=frac{mu_0I}{2s}$$
is the magnetic field at the center of the current-carrying loop.
If $R_2>R_1$, we can assume that the magnetic field passing through the inner loop is uniform and equals to value at the center. Thus the flux
$$Phi=frac{mu_0I}{2s}pi R_1^2$$
That's exactly what has been done (where $s$ is stands for the radius of outer loop). If you want a correct expression you need to find the magnetic field
$$B(r,theta)=  ???$$
which is a non-trivial task.

Magnetic flux with two concentric cirular loops

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