What is the point of maximum magnetic field on the axis of a ring wire?

The equation for magnetic field on the axis of a ring is:

B =

Clearly this is maximum when the denominator has its minimum value, or when
z = 0. Thus the maximum field is simply at the center of the ring.

Problem :

Two rings of radius 1 cm and parallel current I are placed a distance of 2 cm
apart, as shown below. What is the magnitude of the magnetic field at the point
on their common axis midway between the two rings?

The contribution of both rings to the magnetic field is in the positive
direction and, since the point is equidistant from both rings, both contribute
the same magnitude of magnetic field. Thus we simply need to calculate the
contribution by one ring, and double it. The contribution by one ring is given
by:

B = = =

Thus the total magnetic field at that point is:

B = =

Problem :

A semi-infinite solenoid is a solenoid which starts at a point, and is
infinite in length in one direction. What is the strength of the magnetic field
on the axis of the solenoid at the end of a semi-infinite solenoid?

To solve this problem, we use the superposition principle. If we put two semi-
infinite solenoids end to end, we have an infinite solenoid, and the field
strength at any point in the infinite solenoid is . By
symmetry, the contribution of each semi-infinite solenoid is equal, so the
contribution of one semi-infinite solenoid must be exactly one half of the
magnetic field in an infinite solenoid, or

B =

This problem displays the power of the superposition principle, which simplifies
what would be a complex calculation.

Problem :

Two rings, both with radius b, with a common center and the same current I are
placed at right angles to each other, as shown below. What is the magnitude and
direction of the magnetic field at their center?

Each ring contributes the same magnitude of magnetic field, though in
perpendicular directions, as shown below.

The magnitude of each vector is simply:

B =

Since they are at right angles, the magnitude of the resultant vector is simply:

B = = =

The resultant vector points at an angle of to the plane of each
ring, or up and to the right in our figure above.