Section 3.3 · Further exercises · 1 of 10
Exercise 3.3.1: Potential of a ring on its axis
The following exercises use only the relations collected at the beginning of this subsection. Exercises 3.3.1–3.3.4 compute potentials from the charge distribution, Exercises 3.3.5–3.3.7 work from the field or the potential, and Exercises 3.3.8–3.3.10 solve Laplace's and Poisson's equations.
Potential of a ring on its axis
#cf. Home exercise 3.3.2
A circular loop of radius in the -plane, centred at the origin, carries a uniform line charge density . Find the electrostatic potential at and, from it, the electric field on the axis. Compare with Problem 3.1.2.
Adapted from: B. M. Notaroš, Electromagnetics, Example 1.14.
Superposition along the ring: every element is at the same distance from P, so the integral is trivial (Figure 3.3.3b). Then on the axis.
The source point is , so
the same for every element of the ring. With the potential is
On the axis the field is along by symmetry, and
For ,
a point charge . At the field vanishes, as the symmetry of the ring requires.
This is the field of Problem 3.1.2 (with ), found here with a scalar integral whose integrand does not even depend on : the vector cancellations of the direct method are automatic. The potential is largest at the centre, , where the field vanishes. This is a maximum of along the axis (but, as in Home exercise 3.3.3, not in the plane of the ring).