A circular loop of conducting wire of radius a carries current I. Find
the magnetic field on the axis of the loop a distance h from the plane of
the loop by direct integration of the Biot-Savart Law.
If a small circular circuit of radius 
is placed at this position
(so that the magnetic field may be considered uniform over the area of the small
loop) such that the planes of the two circuits are parallel, find the mutual
inductance between them.
DISCLAIMER : Note that the circuit approximation is used implicitly
in the above calculations. The circuit approximation assumes that the
propagation time of electromagnetic signals across a circuit is negligible on the
time scales on which currents change in the circuit. In principle, to say there is
a time-dependent current I(t) which is the same throughout a system is a nono
in electromagnetism. In principle changes in currents take a finite time to propagate
across a circuit. We also assume that when a current changes in a primary circuit
the induced emf is seen immediately in the secondary circuit. Again, in
principle there is a finite propagation time.
It is believed to be true in general in electromagnetism that
as a relation between spatial change of 
and temporal change of 
at a point but in circuit calculations we integrate
this to :
implying that the electric field round the edge of S responds immediately to changes
in far from the edge. This can be an approximation only.
SOLUTION:
The element of magnetic field at distance h along the axis, due to a current element
is:
The components of the various 
along the axis all add, while those normal to the
axis sum to zero. The magnitude of the component of 
along the axis is:
So the total field along the axis is:
The magnetic flux through the loop of radius (normal to ) is:
Since the mutual inductance M is defined by 
:
