Re: Kirchhoff's laws and conservation laws

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Daniel's question:

> I have always seen Kirchhoff's voltage law derived from
> conservation of energy, not flux.  The potential energy
> of a charge is well defined at each point in the circuit.
> As the charge circulates around the circuit, it must
> return to the same potential energy.  Is there a flaw
> in the derivation from conservation of energy?

Kirchoff's voltage law is a consequence of the electric field being
curl-free.  It is only curl free if there is a conservation of
magnetic flux, otherwise the nonzero curl of the E-field is
determined at the negative of the time rate of change of the magnetic
flux density by Faraday's law of magnetic induction.  Once we have
the condition that the E-field is curl free then we have the
existence of a potential function whose gradient gives the negative
of the E-field.  If the magnetic flux is not conserved then there is
*no* such potential voltage function that gives the actual electric
force field, and there is then no electric potential energy to
consider (conserved or otherwise).

> (I agree that Kirchhoff's current law is a direct
> consequence of conservation of charge.)

Actually it is a consequence of a combination of the conservation of
charge *and* a hypothesis of the currents being in a steady-state.
IOW Kirkoff's current law requires that div(j) = 0 and this is only
the case when the d[rho]/dt term in the charge conservation
law (i.e. charge continuity equation) vanishes.

David Bowman