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





-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@mail.phys-l.org] On Behalf Of John
Denker via Phys-l
Sent: Friday, February 16, 2018 1:15 PM
To: Phys-L@Phys-L.org
Cc: John Denker <jsd@av8n.com>
Subject: Re: [Phys-L] Electric Current ... including steady current in a loop

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.

I think I've gotten my answer. Thanks to all who have contributed. It makes total sense to say that the net flow of any conserved (at least under the prevailing conditions) quantity that flows into a closed region of space is equal to the rate of change of the amount of that something in that region of space; that's what we mean we say it's conserved. But to say that one part of the flow of something through a closed surface, or the flow of something through an open surface, is the rate of change in the amount of that something, bothers me; I can live with it though. I was hoping I was missing something.

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