Section 3.2 · Further exercises · 2 of 10
Exercise 3.2.2: Cylinder with a nonuniform 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.
Cylinder with a nonuniform charge
#cf. Home exercise 3.2.3
An infinitely long cylinder of radius carries the volume charge density (), where is a constant. Find inside and outside the cylinder.
Cylindrical symmetry and a coaxial Gaussian cylinder. The enclosed charge is an integral over because the density varies with . Do not forget the Jacobian .
The density depends on only, so the symmetry is that of Problem 3.2.1 and . The left-hand side of Gauss's law is the same as in Exercise 3.2.1, .
The charge inside a coaxial cylinder of radius and length is
Equating both sides,
For the enclosed charge is the whole charge on the length , obtained by setting in the integral: . Hence
Both expressions give at . Outside, the cylinder acts as a line charge . The differential form in cylindrical coordinates confirms the inside field: