A small circular loop of radius $a$ is located at a distance $z$ above the center of a large loop of radius $b(ab,z).$ The planes of the loops are parallel, and their centers are aligned along a common symmetry axis, as shown in the figure below.

$($a$)$ Suppose there is a steady current I in the larger loop. What is the magnetic flux through the smaller loop? $($Assume the area of the smaller loop is small enough that you can consider the magnetic field uniform over it$.)$

$($b$)$ Suppose there is a steady current I in the smaller loop. What is the magnetic flux through the larger loop? $($Assume the smaller loop is small enough that it can be approximated as an ideal dipole.$)$

$($c$)$ Find the mutual inductance and confirm that $M_{21}=M_{12}.($Note: The mutual inductance between objects $1$ and $2-$let's say $M_{21}-$is defined as the ratio of the flux through object $2$ due to the magnetic field generated by the current $I_1$ in object $1.$ That is$,{}_{B2(1)}=M_{21}{I}_1.)$