I wrote:
>> example: lots of "DC" circuits don't
>> make sense at zero frequency. You have to
>> consider the case where some things are
>> slow compared to other things, and then
>> (if necessary) pass to the limit.
On 11/17/2003 10:39 PM, Herbert H Gottlieb wrote:
>
> What do you mean by a "DC" circuit in the above example?
> Obviously it could not be a simple circuit with only
> simple resistors and a battery. Are you referring to
> dynamic dc circuits such as those containing capacitors
> that start to charge and then discharge through other circuit
> elements before they reach saturation?
Well, suppose that circuit (with simple resistors and
a battery) was assembled in the recent past. Or
suppose we add to it a switch that was closed in
the recent past. If you open/close the switch,
it's not really a DC circuit. You can argue that
in the absence of explicit inductors and capacitors
the AC behavior is the same as the DC behavior, or
you could boldly assert that the time of the switch
closure doesn't matter if it was "long enough" ago,
but those are subtle arguments that IMHO implicitly
involve some sort of limiting process such as I
recommended above.
There are always some parasitic inductances along
wires and parasitic capacitances between wires. So,
anything you want to call a "DC" circuit, unless
it has existed unchanged since the dawn of time,
is "DC" only in the sense that the RC times and
L/R times are negligible relative to other timescales
of interest, such as how fast you open/close the
switch, and how closely you scrutinize the results.
=============
Similar remarks apply to the "instantaneous" propagation
of Newtonian gravity and the unbounded speed of propagation
of solutions to the diffusion equation and the Schrödinger
equation. We know that things don't actually propagate
faster than the speed of light, so those equations must
apply only in situations of lengthscale x and timescale
t such that x/t is small compared to c.