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Re: [Phys-l] Current as Vector



James McLean wrote:
I1=2A I2=1A
--------->---------------------->------------
|
V I3=1A
|
If you try to make vectors out of these, I2=(1A)i-hat, I3=-(1A)j-hat, and I2+I3 has a magnitude of 1.414A, which has no useful physical meaning that I can detect. It certainly doesn't make the "junction rule" any easier to handle.

Yesterday I thought this was an excellent point. Now I think
it is mostly just a red herring. In it remains valuable insofar
as it lets me see why people were confused about the vector
current issue.

The diagram and remarks quoted above do not prove that something
is not a vector. For instance, let us re-interpret the diagram
as follows:
I1 = velocity of cars entering a parking lot via the front
I2 = velocity of cars exiting via the back
I3 = velocity of cars exiting via the side.

The three vectors do not "add up". So what? That doesn't prove
that velocity is not a vector.

It remains true that the "junction rule" (i.e. conservation of charge)
involves just adding the scalar currents *with due regard to signs*
but you must not leap to the conclusion that the same conservation
law can be expressed in terms of the vector currents just by adding
the current vectors.

By way of contrast, conservation of momentum can be expressed in
terms of adding the momentum vectors, but that's because it is
really three conservation laws rolled together; each component
of momentum is separately conserved. The conservation law for
vector current is really only one law, involving a tricky combination
of components.

(And as a further contrast, for velocities there is no conservation
law at all. But velocity is still a vector.)

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

Recall that I recognize that the scalar current does exist, and is
often convenient for calculations. However that does not preclude
the existence of the vector current.
1) The fact that the scalar current has no physical interpretation
unless we specify a basis vector (the arrowhead on the circuit diagram)
seems like a dead giveaway to me.
2) The vector current shows up in the Biot-Savart law.
3) The vector current has a particularly simple interpretation in
four dimensions, as part of the [charge,current] four-vector. The
conservation law expresses continuity of the charge world-line, as
discussed (with diagrams) at
http://www.av8n.com/physics/conservative-flow.htm
4) The vector current is equal to the drift velocity times the
charge per unit length. (This is not just dimensional analysis,
this is an exact equality.)