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



On 02/28/2013 03:10 PM, Bruce Sherwood wrote:

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.

This is an interesting and important discussion.

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,

That's true and important.

though that may not often be particularly useful.

It's directly useful in connection with "static electricity"
i.e. high-voltage electrostatics.

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).

The introductory classes can kinda sorta get away with that, insofar
as they restrict attention to elecroSTATICS and magnetoSTATICS.

OTOH there is a genuine dilemma here, because of the risk of
negative transference from statics to dynamics. A lot of things
that are true for statics are just not true for dynamics, and
will have to be unlearned.

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.

Well, yes and no. As physicists and as human beings we search
for unification, but we also search for simplicity. Kirchhoff's
laws are part of a package that also includes *circuit diagrams*
in which the physical geometry of the circuit does not matter;
only the abstract topology matters. This is a profoundly useful
simplification (in cases where you can get away with it).

There are a goodly number of folks who understand things at the
Kirchhoff level, and I'm OK with that. They earn their living
that way. Meanwhile, there is a smaller number of folks who
understand things at the Maxwell level. They get paid a lot
more. They get called in to deal with the high frequency and/or
high precision stuff, among other things.

.... their real nature is obscured by calling them laws.

That's based on a valid observation, but I would prefer to place
the opposite interpretation on the data. I would say that the
nature of these laws sheds a great deal of light on what we
mean by "law". One of the most basic requirements of critical
thinking is to never judge a book by its cover. Just because
we have N different things that are all called "laws" does
*not* mean that they have anything in common. Some "laws" are
remarkably robust, while others are remarkably fragile. The
rules of science require us to keep track of the provisos and
limitations of each rule, on a rule-by-rule basis. Students
resist this, perhaps because it is so much more easily said
than done. Still, it has to be done. Lumping all "laws"
together is a non-starter. Renaming them is not feasible,
and wouldn't solve the problem anyway. The problem is that
you have to learn the provisos and limitations of each rule,
on a rule-by-rule basis. The limitations of Newton's law
of universal gravitation are wildly different from the
limitations of Kirchhoff's laws. That's just how it is.


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.

Actually there are two kinds of breakdown here. A simple diode
is non-Ohmic but Markovian. A light bulb is far worse, insofar
as it is not even Markovian. By that I mean that it has memory.
The voltage at time t is not even a function of the current at
time t, but instead depends on lots of earlier times also.