Section 3.3 · Further exercises · 7 of 10
Exercise 3.3.7: Uniformly charged sphere: potential from the field
About the further exercises of Section 3.3
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.
Uniformly charged sphere: potential from the field
#cf. Home exercise 3.3.7
A sphere of radius carries a uniform volume charge density . Its field (from Gauss's law, Home exercise 3.2.1) is for and for . Find everywhere and verify Poisson's equation inside.
Adapted from: B. M. Notaroš, Electromagnetics, Example 1.18.
Use the recipe "Finding by integrating ": integrate from to infinity, splitting the integral at where the formula of the field changes.
With , by Eq. (3.3.13),
Integrate from to the surface, where the field has the inner form, and add the surface potential:
Inside, with Eq. (3.3.17),
and outside , as it must be in the charge-free region.
Both formulas give at , so is continuous, and for the ball acts like a point charge (Figure 3.3.6a).
The potential is largest at the centre, , and both and are continuous at , since there is no surface charge there.