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*From*: John Denker <jsd@av8n.com>*Date*: Fri, 16 Feb 2018 11:15:20 -0700

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.

**Follow-Ups**:**Re: [Phys-L] Electric Current ... including steady current in a loop***From:*Jeffrey Schnick <JSchnick@Anselm.Edu>

**References**:**[Phys-L] Electric Current***From:*Jeffrey Schnick <JSchnick@Anselm.Edu>

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