To better understand our approach to the role of surface charge in circuits
it may be helpful to describe our highly parallel approach to the role of
surface charge in electrostatics.
We begin E&M in chapter 14 which defines electric field, emphasizes a
strong form of the superposition principle (it's not just that the field is
the vector sum of the contributions of all charges, it is in addition that
the contribution made by any charge is unaffected by the presence of other
charges), and demonstrates the reality of the field by considering
retardation effects. The field of a dipole is calculated as an application
of the superposition principle.
Chapter 15, "Electric Fields and Matter", deals with the polarization of
conductors and insulators caused by applying an electric field, something
that should be considered central to E&M but which is nearly lacking in
many textbooks. We observed that students struggled with the concept that
in equilibrium the field must be zero inside a conductor, because it came
across to them as rote knowledge, and to provide a sense of mechanism we
introduce the notion of a mobile electron sea in metals, and the Drude
model, with the drift speed v = uE, where u is the electron mobility. Apply
a field to a block of metal, and that field drives electrons inside the
metal (very brief transient currents). The direction of these currents is
easily seen to produce surface charge that contributes additional fields of
a kind that weakens the net field in the metal, and thinking iteratively,
the ultimate effect is to drive the net field inside the metal to zero.
(That the polarization charges are on the surface, and that in the final
state the net field indeed vanishes everywhere, are shown to be plausible,
and there is a forward reference to using Gauss's law later to prove these
properties formally.) We replaced a rote rule ("In equilibrium the net
field in a metal is zero everywhere") with a sense of mechanism, that when
fields are applied to conductors surface charge comes into being, in just
such a way as to make the distribution of electric field become zero
everywhere inside the conductor. The emphasis is that the system reacts in
such a way as to create a distribution of surface charge that, in
combination with the applied field, leads to the known final (equilibrium)
state, namely a zero field everywhere in the interior. We show very
approximate diagrams of surface charge on a polarized metal but do not
attempt to calculate the details of the distribution of surface charge.
(Chapter 16 deals with the field of distributed charges of the usual kinds
(rod, ring, disk, capacitor of two finite disks, sphere). Chapter 17 is on
electric potential. Chapter 18 is on magnetic field, and here we define
electron current i = nAv and conventional current I = |q|nAv in the context
of currents making magnetic field.)
Chapter 19 is "Electric Field and Circuits". Exactly as in chapter 15, an
external field (applied in this case by a battery) drives transient
currents (v = uE) that establish a distribution of surface charge which
makes additional contributions to the net field inside the circuit elements
in such a way as to lead quickly to the known final (steady) state, namely
that the distribution of electric field is that which obeys the node rule
(charge conservation in the steady state) and the loop rule (round-trip
path integral of electric field is zero in the absence of a time-varying
magnetic field). We show very approximate diagrams of surface charge but do
not attempt to calculate the details of the distribution of surface charge.
As in chapter 15, the intent is to give a strong sense of mechanism rather
than just stating rules to be followed. (Incidentally, note that during the
brief transient the node rule is not strictly obeyed; the node rule applies
to the final steady state.)
(Chapter 20 is "Circuit Elements", in which we analyze RC circuits
qualitatively in terms of field (with the fringe field of the capacitor
playing an important role), and the rest of the chapter is about the
standard macro analysis of DC and RC circuits, together with developing the
concepts of resistance (making a macro-micro connection between v = uE and
the equivalent macro Ohm's law) and capacitance. The remaining chapters are
on magnetic force, Gauss'a law and Ampere's law, Faraday's law, radiation,
optics, and interference.)