Re: [Phys-L] Electric Current ... including steady current in a loop

[John Denker <jsd@av8n.com>]

On 02/16/2018 06:11 AM, Jeffrey Schnick wrote:

> In the case of a simple circuit such as a resistor connected across a
> battery, the current is typically written as 
> I=dq/dt.			[1]

Executive summary:  The "q" that appears there is
the charge in some region, and the "I" is the
*total* current flowing into that region.

Longer version:  Equation [1] should not be taken
as the definition of current.  It should be thought
of as a slightly simplified conservation equation
aka continuity equation, i.e. conservation of
charge and continuity of current.

It helps to think of charge and current as the
timelike and spacelike components of a single
spacetime vector.  If you have some charge that
just sits there, its worldline (in spacetime) is
parallel to the t-axis.  The spacelike component
i.e. the current is zero.  If the charge is moving,
the worldline is tilted.  In this situation both
the charge and the current components are nonzero.

Let's be clear:  Charge and current are intimately
related ... but I do not think of charge being
defined in terms of current, nor vice versa.  I
think in terms of the spacetime vector.

In particular, equation [1] can be interpreted as
follows:  the charge is related to how much stuff
is crossing the boundaries of a certain cell by
crossing the "early" and "late" boundaries in the
timelike direction, while the current is related
to stuff crossing the lateral sides of the cell,
crossing in the spacelike directions.  Many diagrams
of this sort of thing can be found here:
  https://www.av8n.com/physics/conservation-continuity.htm

A particularly incisive example is steady flow
around a loop.  You can create such a situation
using a time-varying magnetic field, as in a
transformer or betatron.  If you set up a current
in a superconducting loop, it will continue forever;
this is called a /persistent current/.
   https://www.av8n.com/physics/conservation-continuity.htm#fig-conservation-loop

The best way to proceed is to write the conservation
aka continuity equation in differential form:

	∂ρ/∂t + ∇⋅J = 0		[2]

You can then, if you wish, integrate equation [2]
over some region.
   (When doing the integral, use the Stokes theorem
   to write the current as an integral over the
   boundary of the region.)

This results in essentially equation [1].  However,
the derivation makes it clear that the "I" that appears
in the equation is the *total* current flowing into
the region, i.e. the integral of J over the *whole*
boundary (with a minus sign).  So perhaps equation
[1] should be written in terms of I_total or some
such.

It is often useful to define a local current I, not
to be confused with I_total ... but once again we
find that equation [1] is not the definition of I.