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[Phys-L] The status of Kirchhoff''s laws



Recently there was some commentary on the status of Kirchhoff's laws. I'd
like to present a somewhat unconventional take on them, with pedagogical
consequences.

In the absence of time-varying magnetic fields (so I'm talking about DC or
near-DC currents in a circuit, and negligible externally applied
time-varying magnetic fields), curl(E) = 0, which implies that the
round-trip path integral of E must be zero, no matter what path you take.
This is what the "Kirchhoff voltage law" says, but by giving it a name and
constantly referring to it, the student naturally thinks that somehow this
is a special law, which applies only to circuits, and can be applied only
for paths that follow circuit elements. No, integral(E.dl) = 0 is
completely general under DC conditions. The path need not follow circuit
elements -- it can wander outside the wires, though that may not often be
particularly useful. In intro E&M it is typical not to mention curl, but
that the round-trip integral of E is zero follows from the path
independence of this integral in the presence of stationary (or slowing
moving) point charges, whose field ultimately follows from Gauss's law
(differential or integral form).

As for the "KIrchhoff current law", that's just charge conservation in the
steady state, namely that charge into any region must equal charge out if
the charge of the region is not changing. (When you first close the switch,
there is a transient leading to the steady state, and during that very
brief time charges build up on the surfaces of various regions of the
circuit, and the "Kirchhoff current law" does not hold until the steady
state is established.)

The important point is that circuits are not a separate domain of physics
and in particular are not separate from electrostatics. It is a strong
violation of the physicist's search for unification to present circuits as
somehow a subject completely divorced from all other aspects of E&M and
subject to their own special laws.

A less serious issue is "Ohm's law". It is preferable to say that there are
(approximately) ohmic materials (in particular, if the temperature doesn't
change much), and there are definitely non-ohmic materials. "Ohm's law" is
just an approximate description of some but by no means all materials, and
referring to this in terms of a "law" is misleading.

Moreover, the combination of Kirchhoff "laws" and Ohm's "law" may stimulate
students to confuse the two, since their real nature is obscured by calling
them laws. In particular, in the very simple circuit consisting of a
battery and an ohmic resistor, students can be quite confused as to the
status of V = IR, not being clear as to whether this is a statement of
"Kirchhoff's voltage law" or "Ohm's law". I think it's better to say that
because the round-trip integral of E is 0, and we can choose any path, and
we'll choose a path going through the battery, that we get Delta-V_batt +
Delta-V_res = 0. Then from the knowledge that the resistor is
(approximately) ohmic with resistance R, we have Delta-V_res = -RI (if our
path goes in the direction of the current), and that Delta-V_batt is
(approximately) equal to the battery's emf. We then conclude that (emf) +
(-RI) = 0. (I judge it to be important that we always write Delta-V, not V,
in these discussions.)

Finally, this detailed analysis leaves open the possibility of refining the
analysis to say Delta-V_res = emf - rI, where r is the internal resistance
of the battery, in which case we get emf - rI - RI = 0.

I know for certain that as a student, and for too many years thereafter, I
was quite confused by these matters.

Bruce