Why Surface Charges?

Interest in the surface charges on the wires of a circuit has been revitalized by the publication of an introductory calculus-based text by Chabay and Sherwood.[1] They use the concept of surface charges to link the usually disjoint topics of electrostatics and circuits, and to help students develop a conceptual understanding of circuits.

The typical second-semester physics course covers, among other topics, electrostatics and electric circuits. The student is told ``current is charges in motion'', which is not an obvious statement.

Moreover, the student is taught how to calculate current and potential differences in single- and multi-loop DC circuits, but typically she does not ask or answer questions such as

• How does the battery ``know'' how much current to send to the circuit?
• Why does the brightness of a light bulb not change as the bulb is moved towards and away from the only (apparently) source of energy, charge, and field?
• Why, in a resistor-capacitor circuit, do the charges on a capacitor plate move away from the oppositely-charged plate and travel around the circuit? Don't opposites attract?

These questions are best answered by analyzing the surface charges on the wires of the circuit and the feedback mechanism that adjusts them very quickly. For a wire of uniform resistivity, there can be no free charges in the bulk, but only on the surface. The creation of the surface charges, and the feedback process, will be illustrated below.

Chabay and Sherwood use qualitative diagrams of surface charges in their text, in part because quantitative diagrams have not been available. Only a few circuits are amenable to analytic solutions, and they don't resemble real circuits: infinite straight wires, coaxial wires and batteries, spherical batteries in an infinite resistive medium, etc.

I have recently computed the distribution and motion of free charges for a series of circuits similar to some of the examples in their text. The calculations use Coulomb's law and the simple relationship between current density and electric field,

$$\vec{J} = \sigma\vec{E}$$ (1)

where $\sigma$ is the conductivity. The calculation was done by a relaxation technique: the electric field due to an initial configuration of charges was computed on a mesh of points in the wires, charges were moved according to the field and conductivity, and then the field was re-computed. Eventually a self-consistent set of charges and fields was determined, where the charges create the electric field that move the charges to their positions. The circuits do not contain batteries, but large capacitors, to avoid the difficulties of having to deal with non-conservative electric fields.

It is tempting to view the sequence of pictures (see Fig. 2) that result from the relaxation calculation as the time-response of the circuit. Unfortunately, that is not correct: the calculations assume the speed of light is infinite, and so changes in charges there affect the electric field here instantly. The sequence of pictures does help isolate specific features, however, and is of pedagogic interest for that reason.