**Problem : **

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:

Clearly this is maximum when the denominator has its minimum value, or when

**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?

Two rings with a common axis. What is the strength of the field at point
*P*
?

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:

Thus the total magnetic field at that point is:

**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

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?

Two rings at right angles to each other. What is the field at point
**
***P*
?

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

The two contributions to the magnetic field of problem 5

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

The resultant vector points at an angle of

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