W9_parallel_resonance.eps

114 Week 3: Potential Energy and Potential

Example 3.4.3: A ring of charge

dθ

z r

a

dl = a dθ

λ

x

y

Figure 30: A ring of charge in thexy-plane, concentric with thez-axis.

Suppose you are given a ring of charge with charge per unit lengthλand radiusaon

thexy-plane concentric with thez-axis. Find the potential at an arbitrary point on the

zaxis.

Although there is a quick and easy answer to this problem (that will beapparent at the end, if not
at the beginning) we will work through this problem in detail to illustrate the general methodology
of finding a potential by integrating over a continuous distribution of charge. The steps are:

a) In suitable coordinates, define a differential “chunk” of the charge. In this problem, that would

be a differential-size arc segment of the ring.

b) Determine the differential charge of the chunk as “the charge of the chunk is the charge per

unit whatever times the differential whatever of the chunk” where‘whatever’ might be length,

area or volume (in this case length).

c) Write a simple expression in suitable coordinates for the differentialpotentialproduced at the

point of interest by the differential (point-like) chunk of charge:

dV=kedq

r

whereris the distance from the chunk to the point of observation. Note well that this is a

scalarintegral, making it relatively simple!

d) Integrate both sides. The left hand side becomesV(~r) at the point of observation (in suitable

coordinates). The right hand side becomes the algebraic expression of the potential (the

answer).

e) Simplify, if appropriate or required.

f) If one wishes to find the field from the potential, remember e.g.

Ez=−dV

dz

Beware L’Hopital’s Rule! That is, if differentiating someplace that the function itself vanishes

(or its functional dependence on certain coordinates vanishes) be sure that you differentiate

at a general pointnearthe limit point andthentake the limit!