Is current flow the same as current flux in electromagnetics?

[Cole A.]

I'm reading an introductory E&M textbook (with an electrochemical emphasis, so charge carriers are ions rather than electrons), and the wording my book uses to describe current is giving me a great deal of confusion that I was hoping someone could help me resolve. My understanding is as follows:

charge: $q,$ in units of Coulombs (C).

current: the rate of flow of charge; $i = \frac{dq}{dt}$, in units of C/s.

current density (or current flux): the rate of flow of charge per unit area; $J = lim_{A \rightarrow 0} \frac{i}{A},$ where the limit converges about a point, in units of C/s/area.

My confusion is when the book uses phrases such as: "Current flows from the positive terminal to the negative terminal..." and "The current flow through the membrane..." I don't understand how a time rate of change (i.e., current) can "flow" anywhere at all, when it is the charge carriers that are physically moving. In my eyes, it is sort of like saying that automobile velocity flows through the tunnel --- but the velocity isn't flowing through the tunnel, the automobiles are. It seems like the correct phrasing should be "charge flow" in both cases. Am I mistaken?

Or, another possibility I'm worried about is that I'm misinterpreting "current flow" in the sense that it really means "current flux." When people say "current flow," are they actually referring to current flux? Are current flow and current flux the same thing?

I guess my question is: if someone were to say, "current flows from the positive terminal to the negative terminal," should this (1) be viewed as sloppy wording and interpreted actually to mean that positive charge flows from the positive terminal to the negative terminal, or (2) be interpreted to mean that a current flux exists that is directed from the positive terminal to the negative terminal? Because viewing it literally as current flow seems meaningless to me.

Thanks

 

[Philip Wood]

You are absolutely right. It is illogical to talk about current flowing. Flows of charge don't themselves flow (except in unusual situations). I've tried to train myself to say "There is a current in the circuit.", when some would say "A current is flowing in the circuit." Yet, only a week ago, I fell into the trap!

For a somewhat similar reason, I've always been wary of talk of time flowing. With respect to what does it flow?

 

[vanhees71]

It's a very bad habit to talk about the "direction of a current". The current is a scalar density and has thus no direction.

Current density has a direction. Within classical physics (on the level of a hydrodynamical picture for a plasma or a gas of conduction electrons in a conductor, ions in solutions, etc) and in the non-relativistic limit which is applicable in nearly any every-day application of classical electromagnetics (an important exception are homopolar generators, which cannot be described non-relativistically, but that's not our topic here) it's given by
$$\vec{j}(t,\vec{x})=\sum_{j} q_j n_j(t,\vec{x}) \vec{v}_j(t,\vec{x}),$$
where $q_j$ is the charge of the particle (species $j$), $n_j$ the number density, and $\vec{v}_j$ the velocity field.

Now, from this fundamental quantity, the current is derived as the amount of charge flowing through a given area per unit time. The meaning of the sign of the current is thus simply given by the choice of orientation of this area:
$$i(t)=\int_{A} \mathrm{d}^2 \vec{A} \cdot \vec{j}(t,\vec{x}).$$
That's all. There's no need to get confused with "technical direction of current" vs. particle flow, etc. All signs are properly taken into account here, and the meaning of the sign of the current is whether there is net flow in direction of the orientation of the area ($i>0$ or opposite to it ($i<0$).

 

[Mesmer8]

There is a page on electricity somewhere online written by a former teacher which mentions the awkwardness of the sentence, "Current is flowing (through a circuit)." It is merely a common error in parlance and print to be understood as meaning, "charge flows..."

Indeed, the electron does not like to go towards the negative terminal. On its own, it will prefer the positive terminal, so only electron "holes" are what, like Spaghetti Western wagon wheels, appear to move from the positive terminal (i.e., the terminal which is less negative than the officially-negative terminal) to the negative. The holes going towards the supply of electrons are merely ghosts of formerly closed valence shells falling, "like lightning," back to Earth. But nothing is actually falling - only the holes, or gaps, of the erstwhile valence electrons fall, which are, themselves, being displaced towards the "positive" terminal, or electrode. Whereas the flow of electrons is no illusion, it's weirder to talk about holes moving.

On a similar note, it is often said that the ion is what deposits onto the cathode in the aqueous solution of the electroforming tank. In fact, what actually deposits is a closed, normal metal which the ion couldn't resist becoming. Once deposited, it has ceased to be an ion. Before ceasing to be an ion, it has not deposited - only approached. Ions, therefore, are thrown at the cathode, but it is "grounded" metal which deposits on the mandrel and the growing electroform.- Mesmer8

 

[Dadface]

I think there is a confusion here between current defined as a flow of charge and magnitude of current defined as a rate of flow of charge. If current is defined as a flow of charge,as I think it usually is, then it is meaningful to talk about its direction.

 

[Mesmer8]

I agree, Dadface. All's I meant is that not until we know which charge carrier we are talking about can we have a fair guess about the direction of the flow. It is redundant as the OP discovered to say that the "current flows*" if all one means is that the charge carriers flow. Their direction will describe the circuit. Holes left behind appear to travel to the more negative terminal. But the only reason we can say that there's a positive terminal is that it wants "free" electrons. The waterfall of holes from the less negative potential to the more negative is the complement to the migration of the bumped valence electrons which are keen on reducing the Voltaic pressure, not unlike hot vapor in a chamber.

I recently opined that lust is merely scalar whereas love is a vector.


- Mesmer8


* The internet post by the aforementioned teacher likened saying "charge flows" to saying "flow flows." It does, but it's an awkwardism. The river water is what truly flows, and, as proof, we can detect the presence of a watery current therein.

 

[Philip Wood]

Van Hees

Many thanks for this clear summary.

Surely one can talk about the direction of a current, meaning its sense around a circuit. Suppose we have a square circuit NESW (N = North etc.) laid out on the ground, and that we've chosen an area of cross-section of wire (say in the section NE) whose normal is pointing from N to E. If we calculate the current as you specify, and find that it turns out to be positive, we can surely say that the sense of the current is NESWN (clockwise from above)?

 

[vanhees71]

To stress it again! The current $i$ is a scalar quantity. It's sign gives the direction of the net-charge flow through an oriented area, i.e., an area with vectorial area elements in a certain well defined direction.

You can avoid a lot of confusion, e.g., in the theory of quasistationary circuits (AC), particularly when you take into account inductivity and Faraday's Law in integrated form. There you have to specify the orientation of the boundary curve positively relative to the orientation of the enclosed area, when integrating Faraday's Law,
$$\vec{\nabla} \times \vec{E}=-\vec{B}.$$
For details, see my Texan freshmen lectures on electricity and magnetism:

http://fias.uni-frankfurt.de/~hees/physics208/phys208-notes-III.pdf

p. 102

See also the examples for calculations of AC circuits on the corresponding website:

http://fias.uni-frankfurt.de/~hees/physics208.html

The current density $\vec{j}$ is the corresponding local quantity. In the more general language of theoretical physics it's the flow of electric charge. You don't need to care about the sign of the charge of the microscopic carriers. It's very important to keep this in mind, particularly in relativistic physics, where electric charge is a conserved quantity, while particle numbers are usually not. Only particle-number like quantities as net-baryon number (i.e. the number of baryons minus the number of anti-baryons) is conserved, and only for such conserved quantities the total "charge" is a scalar quantity, while the four-current vector $j^{\mu}=(\rho,\vec{j})$ is always a four-vector field.

 

[Cole A.]

Thank you for the responses, everyone.

 

[Philip Wood]

Van Hees. To get things clear: I'm not trying to assign a direction to current as to a vector. But I do want to assign a sense (one of two alternatives) to a current around a circuit of wires. The assignment would be made according to (1) the chosen direction of the normal to a cross-section of the wire, (2) whether the current, calculated as the integral over the cross-section of current density, is positive or negative. But even this, you imply, is wrong?

 

[vanhees71]

No, that's not wrong, but precisely what I tried to day! Just define

$$i(t)=\int_A \mathrm{d}^2 \vec{A} \cdot \vec{j}(t,\vec{x}).$$

Here you have to (arbitrarily!) choose an orientation of the area. Then the sign of $i$ gives the net flux through the area relative to this chosen direction of the surface-normal vectors. That's all, and that makes everything well-defined and unique.

 

[Philip Wood]

Sorry for the misunderstanding. I learn a lot from your posts.

 

[Dadface]

I think this discussion has moved away from the original question where in the main ColeA was referring to phrases used in his book.
He seems to have answered his own question.When books use phrases such as "the current flows from one pole to the other they should make it clear that it is the charges that flow" and not the rate of change of charge.
The word current is often used rather loosely as I indicated above.

 

[vanhees71]

Right! The more important it is to clarify these things in the very beginning. Only because there are bad textbooks out there one should not justify this sloppy use of notions. Science is to a large extent the clarification of the basic notions of the models used to understand nature!

 

[nasu]

The word "current" is used here with two meanings.
Something flowing is described sometimes as a current: water currents, air currents.
The ocean currents means a directional flow of water and we can talk about their direction.
A flow of charge can be called a current in this sense. It's neither a scalar nor a vector. Just a description of a phenomenon. Like "flight" or "motion" and so on.
We can say that something moves from South to North. The speed is a scalar quantity describing the motion. And velocity is a vector quantity describing the motion.
In other languages there is only one world for both speed and velocity. You have to explicitly say which one you mean.

The quantity that describes the rate of charge flow in electrical current was initially called "intensity of the current". So the usual symbol I. Probably it was introduced initially in French.
So dQ/dt it's technicaly the intensity of the current and just by simplification it is now called "current" and it may lead to confusions.

It is like using "motion" to describe both the phenomenon of change in position and the rate of change in position. Someone could have called it "rate of motion" rather than speed and then the "rate of" would have been dropped. And now some people will discuss if motion can have a direction or not.