Section 3.3 · Further exercises · 6 of 10
Exercise 3.3.6: Is it electrostatic? Potential, charge and work
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.
Is it electrostatic? Potential, charge and work
#cf. Home exercises 3.3.5–3.3.7
Consider the field .
(a) Show that it can be an electrostatic field.
(b) Find its potential with .
(c) Find the volume charge density that produces it.
(d) Find the work done in carrying a charge from the origin to .
Follow the recipe "Is a given field electrostatic?": curl test and component-by-component integration of . Then and .
The first fundamental equation requires :
From , integrating one component at a time:
gives :
From Gauss's law in differential form,
(Equivalently, and .)
The work is independent of the path:
, , : the potential gives back the field.
A negative means that, for , the field itself does the work: the charge moves to a lower potential. Note that the field is not zero where (the plane ): fixes only the local sources, not the field.