26
The Electric Field
Recommended class days:
3 minimum, 4 preferred
Background Information
Chapter 25 introduced the field model, and Chapter 26 will now develop the quantitative analysis
of electric fields and of the motion of charged particles in the field. The goal is not for students
to become experts at calculating fields, although they should be able to do calculations for simple
charge distributions, but for them to recognize standard electric fields such as those of dipoles
and of lines and planes of charge.
Field calculations are based directly upon the principle of superposition. Superposition is a
straightforward and readily understood procedure, whereas, as will be discussed more in
Chapter 27, research has found that only a very small fraction of students are able to understand
Gauss’s law. Further, superposition helps students with their conceptual difficulties by explicitly
linking field calculations to Coulomb’s law.
With one exception, all the standard fields can easily be found using superposition. The
exception is the field of a sphere of charge, for which the integrations—although doable—are too
complex for an introductory class. However, it is quite believable, without a rigorous proof, that
the exterior field of a sphere of charge is the same as that of a point charge. We routinely make this
same assertion, without proof, for Newton’s law of gravity.
At this point in the course, students have been doing calculus long enough that nearly all can
carry out an integration. However, students are still very inexperienced at
using
calculus to analyze
a problem. Their difficulty with this chapter—and it is a major difficulty for most—is not with
doing integrals but with knowing
what to integrate
. Students are completely unfamiliar with the
procedure of dividing a charge distribution into small pieces, writing the superposition of fields as
a sum, and then converting the sum to an integral by using the charge density. The emphasis in this
chapter needs to be on the analysis and the procedures that lead to an integration.
Students also have a hard time visualizing fields in three dimensions. The fields of interest in
this course usually have an axis of symmetry, so you need to encourage students to imagine the
two-dimensional drawings of the textbook, or that you draw on the board, as being rotated about
that axis. Several software packages can provide useful classroom demonstrations giving three-
dimensional renderings of fields.
It is important to explain clearly what the standard two-dimensional
drawings are meant to represent. For example, a parallel-plate capacitor is
usually drawn as shown in the figure. Many students don’t recognize, unless
you point it out, that the drawing shown here represents two
planes
of charge
extending above and below the plane of the drawing. These students interpret the drawing literally
as a two-dimensional field of two short line segments of charge.
The parallel-plate capacitor encounters another common student misconception: that one layer