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Re: [Phys-L] flux and flow rate (was electric current)



I think your definitions generalize nicely to mass flux and mass flow rate. How about heat flux and flow rate. I guess ρ would be the density of energy that is on the move with v being the drift velocity of the energy.
________________________________________
From: Phys-l [phys-l-bounces@mail.phys-l.org] on behalf of John Denker via Phys-l [phys-l@mail.phys-l.org]
Sent: Friday, February 16, 2018 7:50 PM
To: Phys-L@Phys-L.org
Cc: John Denker
Subject: Re: [Phys-L] definition of current

Executive summary: There is an obvious follow-up
question that needs to be addressed: How *should*
we define current? This is a seriously important
fundamental question.

I suggest something really simple: If some stuff has
charge density ρ and is moving with velocity v, then
the current density is
J = ρv [0a]

As applied to a wire, we can integrate over the cross
sectional area of a wire. Then
I = λv [0b]
where I is the current and λ is the charge per unit
length of the moving stuff. This is nicely consistent
with the idea of current being the amount of charge
passing a given point per unit time.

How did we get here? Let's review. This started
with a discussion of the equation:
I_total = dq_total/dt [1]
which applies to some region; I_total is the current
flowing in across the boundary of the region, and
q_total is the charge inside the region.

Equation [1] if you read it left-to-right expresses
continuity of current; if you read it right-to-left
it expresses conservation of charge.

More-or-less equivalently we can write
∂ρ/∂t + ∇⋅J = 0 [2]
where ρ is the charge density and J is the current
density.

Neither equation [1] nor [2] defines current. This
becomes obvious if you consider a persistent current
in a loop of wire, for which ∂ρ/∂t is zero everywhere
and ∇⋅J is zero everywhere, so equations [1] and [2]
are trivially satisfied /without/ telling us anything
about the current. If you integrate equation [2] the
persistent current shows up as an undetermined constant
of integration.

Very hypothetically, I reckon that if charge (or charge
density) were the only thing we could observe, the
persistent current would be unobservable, and nobody
would care how we defined the concept of current.
However, in reality we do care, because the current
couples directly to the magnetic field. That makes
it observable.

Furthermore, special relativity guarantees that since
charge produces an electric field in its rest frame,
it *must* produce a magnetic field in a frame where
it is moving.
https://www.av8n.com/physics/straight-wire.htm

Equation [0] is correct to first order in v/c. If you
want to get fancy, you can write the [charge,current]
spacetime vector as
J = ρ_0 u [3]
where u is the spacetime velocity of the charged
stuff and ρ_0 is the charge density in the rest
frame of the stuff.

If there are multiple kinds of charged stuff, you
have to treat each one separately. The total
charge is
ρ = ρ_a + ρ_b
and the total current is
J = ρ_a v_a + ρ_b v_b
which means there are lots of ways you can have
zero ρ but nonzero J.

Bottom line: Equation [1] is *not* the definition
of charge. Sometimes it can be used to infer the
charge, but sometimes not. You're better off using
equation [0] or equation [3].

=====================

BTW this fundamental issue is not restricted to
electrical current. The same questions arise in
connection with the current i.e. flow of a liquid
flowing around and around in a closed loop of pipe.
The density is unchanging in time everywhere, and
if density were the only thing we could observe I
don't think it would be possible (or necessary)
to define the current. However, in reality there
is momentum associated with the flow, and we can
measure that, e.g. by wiggling the loop and looking
for the gyroscopic precession.
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