[Phys-L] conservation, continuity, and constancy .... was: more about basic DC circuits

[John Denker <jsd@av8n.com>]

A while back there was some discussion of textbooks that propagate
the disastrous misconception that electrical current is somehow
«conserved».

On 12/16/2013 11:31 AM, Philip Keller wrote:
> Rather than "conservation" of current, I would have
> chosen "continuity" 

That is /exactly/ the right word.

If we adopt the spacetime point of view, as we should, then
conservation is exactly synonymous with continuity of world lines.

This can be expressed in terms of the four-dimensional divergence,
i.e. the continuity equation:

  (∂/∂t) J_t + (∂/∂x) J_x + (∂/∂y) J_y + (∂/∂z) J_z = 0        "conservation"
                                                          i.e. 4D continuity


The  corresponding 3D continuity equation applies in special cases 
(magnetic field lines, vortex lines).  It also serves as an approximation 
in cases where you can neglect the timelike term in the 4D continuity 
equation.

Let's be clear:  3D continuity is an amputated part of the conservation
equation, applicable when there is no accumulation.

Similarly:  The idea of /constancy/ is the other amputated part of the
conservation equation, applicable when there is no flow.

Those two parts /working together/ make up the 4D continuity equation,
which expresses conservation.

Bottom line:  Rather than talking about «conservation» of current, which
would be crazy wrong, talk about
  *) conservation of charge
  *) continuity of current
    -- approximate continuity of the 3D current
    -- exact continuity of the spacetime [charge,current] 4-vector

I renamed and greatly revised one of my documents to address this point.
  http://www.av8n.com/physics/conservation-continuity.htm

This is not my best work, but it more-or-less gets across the correct
fundamental ideas.