What is Electric Current? -2

What is Electric Current? - Continued

Wiki (20-Jan-2012) explains Electric Current as

"
For a steady flow of charge through a surface, the current I (in amperes) can be calculated with the following equation:

where Q is the electric charge transferred through the surface over a time t. If Q and t are measured in coulombs and seconds respectively, I is in amperes.
More generally, electric current can be represented as the rate at which charge flows through a given surface as:

"

This is very common explanation. But how many people understand the meaning of this? It seems that we must memorize these equations without understanding what these mean or even without being able to visualize it.

Without understanding the first equation

it is further more difficult to understand its generalization.

This explanation is like one for those who already know well the meaning of Electric Current.

One more

Hyperphiscs' (20-Jan-2012) explanation

"

Electric Current

Electric current is the rate of charge flow past a given point in an electric circuit, measured in Coulombs/second which is named Amperes.

"

Again how many people understand the meaning of "the rate of charge of flow past a given point in an electric circuit" ? This time "past a given point" not "through a surface".

Wiki explains Electric Current in another way (aspect) in Drift Speed (the same site as the above).

"  
Drift speed 

The mobile charged particles within a conductor move constantly in random directions, like the particles of a gas. In order for there to be a net flow of charge, the particles must also move together with an average drift rate. Electrons are the charge carriers in metals and they follow an erratic path, bouncing from atom to atom, but generally drifting in the opposite direction of the electric field. The speed at which they drift can be calculated from the equation:

or v = I/nAQ

where

I is the electric current

n is number of charged particles per unit volume (or charge carrier density)

A is the cross-sectional area of the conductor

v is the drift velocity, and

Q is the charge on each particle.

"

In this, I is the total quantity of charges have gone through the cross-sectional area of the conductor during the time (t) with a velocity of v (m/t), which means the total quantity of charges in a volume created by the the cross-sectional area (A) and the distance v x t ((m/t) x t). This is easier to understand because this is a fact or an observation of a fact - generalization is not require.

Generalization processes1) First generalization

Hyperphisics says " the rate of charge of flow past a given point in an electric circuit". To get this we must divide the both side by / m³

I / m³ = nAvQ / m³ = nQAv / m³
As A in ＜m²＞ and v in ＜m/t＞ in unit
nQAv / m³ = nQA ＜m²＞v ＜m/t＞ / m³ = nQ/t
nQ is the total Q (Charges) and change it Qtotal. Then we can get I /m³ = Qtotal/t
But what does the right hand side I /m³ means? Not I but I /m³.
This is the first generalization. I /m³ is the density of I (total current passed), or we can say an average of I ( the flow of charges past a given point). We also call it the rate of charge of flow past a given point or the density of current in terms of time (the time dimension in stead of the spatial dimensions as a point is zero dimension at least mathematically).

Similarly we can think of the average I through a surface by dividing the both sides of the equation by m².
I / m² = nAvQ /m² = nQAv / m²
nQAv / m² = nQA＜m²＞v ＜m/t＞ / m² = nQm/t =  (also = nQv)
This time the right hand side is not nQ/t but nQm/t or nQv (where v is velocity). We do not elaborate this here.
nQAv / m² = nQm/t
As far as the rate or the average are concerned (average over area versus (nQ/t) x distance)
nQAv = nQ/t ---- Generalized I = nQ/t = Qtotal/t

Again similarly we can think of the average I through a line (or maybe more precisely on or in a line) by dividing the both side of the equation by m.
I / m = nAvQ /m = nQAv / m
nQAv / m = nQA＜m²＞v ＜m/t＞ / m = nQm²/t
This time the right hand side is nQm²/t. What does m²/t means? We do not elaborate it here. 
nQAv / m = nQm²/t
As far as the rate or the average are concerned (average over line versus (nQ/t) x area)
nQAv = nQm²/t ---- Generalized I = nQ/t = Qtotal/t

The above calculations seem redundant. " I = Q/t " is a definition.

2) Second generalization

As we quoted above, Wiki says

More generally, electric current can be represented as the rate at which charge flows through a given surface as:

As

a steady flow of charge through a surface

we must generalize this under a more generalized a changing flow. Steady state is included in changing state. Steady state is a special case of changing state when the rate of change is zero. So

and in a more general way, not "charge flows through a given surface" (Wiki) but " the rate of charge of flow past a given point" (Hyperphysics). Quantity at an absolute point could be averaged by spatial dimensions but could be averaged by the time dimension (t).

This reminds me of Divergence (Theory) but Divergence does treat a flow not involve time - so a kind of static flow, then leads to the density of something in a flow or rather flux form.



sptt