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[Phys-L] conservation of charge (and continuity of current) WITHOUT constant density



Hi Folks --

I made a video to demonstrate flow of charged particles. They cross the
boundary into our field of view at the left, flow from left to right,
and then exit our field of view. At no point are any particles created
or destroyed. There is strict conservation of charge and continuity of
current.

Furthermore, this is a steady flow – not just conserved, but also steady.
Steady flow means the amount of charge crossing any given contour of
constant x is the same, independent of time, for all times between the
beginning and end of the movie. (If you imagine that the end of the
movie portrays the charges coming to a stop, that is an example of
unsteady flow. Charge is still conserved, but the flow is unsteady.)

However (!) this does not mean that charge behaves like an
incompressible fluid. We have conservation of charge, continuity
of current, and steady flow, but all of that together is not
sufficient to guarantee that the density of charge remains constant.

Situations like what you see in the video happen all the time in the
real world. For example, this happens whenever you make a junction
between two different types of metal; the carrier density is likely
to be substantially different in the two metals. The same is true to
an even more extreme degree at the junction between a metal and a
semiconductor.

Again: Please do not imagine that charge behaves like an incompressible
fluid. A lot of people who ought to know better get this wrong. The fact
is, in the video, the density in the middle is a factor of two less than
the density near the ends.

In particular, if you want Kirchhoff’s laws to be reliable, you need
to restrict attention to the DC limit, i.e. to steady situations. In
such a situation, the charge density at any particular point is constant
because charge is conserved and the flow is steady ... !not! because the
fluid is allegedly incompressible.

https://www.av8n.com/physics/conservation-continuity.htm#sec-speedup

The programs used to create the movie are included.