Section 3.2 · Further exercises · 7 of 10
Exercise 3.2.7: From the field to the charge
About the further exercises of Section 3.2
Ten further exercises in the spirit of Problems 3.2.1–3.2.5 and the home exercises. All of them use Gauss's law in its integral or differential form,
together with symmetry and, where useful, superposition. Exercises 3.2.1–3.2.2 have cylindrical symmetry, Exercises 3.2.3–3.2.4 planar symmetry, and Exercises 3.2.5–3.2.9 spherical symmetry (or a symmetry argument of another kind), while Exercise 3.2.10 checks the law for a field without any symmetry.
From the field to the charge
#cf. Home exercise 3.2.2
Inside a sphere of radius the electric field is ( constant), and there is no charge outside. (a) Find the volume charge density using the differential form of Gauss's law. (b) Find the total charge in two ways. (c) Find outside.
Run Gauss's law backwards: the differential form gives the density from the field, and the integral form on the sphere gives the total charge directly. The two must agree.
Integrating the density,
and from the integral form on the sphere , where ,
Outside, the enclosed charge is on every Gaussian sphere:
The two forms of Gauss's law are the same statement, related by the divergence theorem (Section 2.4), and they give the same total charge. The field is continuous at ( from both sides). The density found here has the form of Home exercise 3.2.2 with , and indeed Home exercise 3.2.2 gives .