When I thought about this, I thought along the lines of Clarence Bennett's
response that the switch contacts also constitute a miniature capacitor.
I imagined one part of the Gaussian surface goes between the capacitor
plates as Carl Mungan originally described. But I also imagined another
portion of the Gaussian surface goes right through the switch contacts. If
the switch is opened before the capacitor has charged to the full battery
potential difference, then examination of the charging circuit will show a
potential difference across the battery, a potential difference across the
capacitor, and a potential difference across the switch contacts (another
capacitor). The potential differences across the capacitor and switch have
to add to equal the potential difference across the battery according to the
loop theorem.
When the switch is closed and the charges in the wires take on a new
distribution, I assume Gauss' law is still valid, but more difficult to
evaluate. Indeed, it is more difficult to evaluate in the switch-open case
if we don't run the Gaussian surface through the switch contacts.
At any rate, constructing the Gaussian surface through the switch contacts
helps show that the flux across the portion of the surface inside the
original capacitor is not the only flux for this Gaussian surface. Once we
realize there are other non-zero flux areas of the surface we can at least
imagine that Gauss' Law survives even though exact evaluation may be
difficult.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817