Electric Current
Discussion
definitions
Electric current is defined as the rate at which charge flows through a surface (the cross section of a wire, for example). Despite referring to many different things, the word current is often used by itself instead of the longer, more formal "electric current". The adjective "electric" is implied by the context of the situation being described. The phrase "current through a toaster" surely refers to the flow of electrons through the heating element and not the flow of slices of bread through the slots.
As with all quantities defined as a rate, there are two ways to write the definition of electric current — average current for those who claim ignorance of calculus…
I =  Δq 
Δt 
and instantaneous current for those with no fear of calculus…
I = 

Δq  =  dq  
Δt  dt 
The unit of current is the ampère [A], which is named for the French scientist AndréMarie Ampère (1775–1836). In written languages without accented letters (namely English) it has become customary to write the unit as ampere and, in informal communication, to shorten the word to amp. I have no problem with either of these spellings. Just don't use a capital "A" at the beginning. Ampère refers to a physicist, while ampère (or ampere or amp) refers to a unit.
Since charge is measured in coulombs and time is measured in seconds, an ampère is the same as a coulomb per second.
⎡ ⎣ 
A =  C  ⎤ ⎦ 
s 
This is an algebraic relation, not a definition. The ampère is a fundamental unit in the International System. Other units are derived from it. Fundamental units are themselves defined by experiment. In the case of the ampère, the experiment is electromagnetic in nature.
The ampère is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular crosssection, and placed one meter apart in vacuum, would produce between these conductors a force equal to 2 × 10^{−7} newton per meter of length.
This means that the coulomb is defined as the amount of charge that passes through a surface when a current of one ampère flows for one second.
[C = A s]
When I visualize current, I see things moving. I see them moving in a direction. I see a vector. I see the wrong thing. Current is not a vector quantity, despite my welldeveloped sense of scientific intuition. Current is a scalar. And the reason is… because it is.
But wait, it gets weirder. The ratio of current to area for a given surface is known as the current density.
J =  I 
A 
The unit of current density is the ampère per square meter, which has no special name.
⎡ ⎣ 
A  =  A  ⎤ ⎦ 
m^{2}  m^{2} 
Despite being the ratio of two scalar quantities, current density is a vector. And the reason is, because it is.
Well… actually, it's because current density is defined as the product of charge density and velocity for any location in space…
J = ρv
The two equations are equivalent in magnitude as shown below.
J =  ρ  v  
J =  q  ds  =  s  dq  =  1  I  
V  dt  sA  dt  A  
J =  I  
A 
Something else to consider.
I = JA = ρvA
Readers familiar with fluid mechanics might recognize the right side of this equation if it was written a bit differently.
I = ρAv
This product is the quantity that stays constant in the mass continuity equation.
ρ_{1}A_{1}v_{1} = ρ_{2}A_{2}v_{2}
The exact same expression applies to electric current with the symbol ρ changing meaning between contexts. In fluid mechanics ρ stands for mass density, while in electric current it represents charge density.
microscopic description
Current is the flow of charged particles. They are discrete entities, which means they can be counted.
n = N/V
Δq = nqV
V = Ad = AvΔt
I =  Δq  =  nqAvΔt 
Δt  Δt 
I = nqAv
A similar expression can be written for current density. The derivation starts off in scalar form, but the final expression uses vectors.
J =  I  =  nqAv 
A  A 
J = nqv
solids
conduction vs. valence electrons, conductors vs. insulators
Drift motion superimposed on thermal motion
Bridge text.
The thermal speed of the electrons in a wire is quite high and varies randomly due to atomic collisions. Since the changes are chaotic the velocity averages out to zero.
When a wire is placed in an electric field, the free electrons accelerate uniformly in the intervals between collisions. These periods of acceleration raise the average velocity above zero. (The effect has been greatly exaggerated in this diagram.)
thermal velocity of an electron in copper at room temperature (classical approximation)…





fermi velocity of an electron in copper (quantum value)…





drift velocity of an electron in 10 m of copper wire connected to a 12 V car battery at room temperature (mean free time between collisions at room temperature τ = 3 × 10^{−14} s)…







The thermal velocity is several orders of magnitude greater than the drift velocity in a typical wire. Time to complete the circuit is about an hour.
liquids
ions, electrolytes
gases
ions, plasma
 2:02 PM  Transmission line disconnects in southwestern Ohio
4. Stuart  Atlanta 345 kV
This line is part of the transmission pathway from southwestern Ohio into northern Ohio. It disconnected from the system due to a brush fire under a portion of the line. Hot gases from a fire can ionize the air above a transmission line, causing the air to conduct electricity and shortcircuit the conductors.
Source
historical
The symbol I was chosen to represent the intensity of a current by the 19th century French physicist and mathematician AndréMarie Ampère.
Pour exprimer en nombre l'intensité d'un courant quelconque, on concevra qu'on ait choisi un autre courant arbitraire pour terme de comparaison…. Désignant donc par i et i' les rapports des intensités des deux courants donnés à l'intensité du courant pris pour unite…. To express the intensity of a current as a number, suppose that another arbitrary current is chosen for comparison…. Let us use i and i' for the ratios of the intensities of two given currents to the intensity of the reference current taken as unity…. AndréMarie Ampère, 1826 AndréMarie Ampère, 1826 (paid link)
The term intensity now has an unrelated meaning in physics. Current is the rate at which charge flows through a surface of any size — like the terminals of a battery or the prongs of an electrical plug. Intensity is the average power per unit area transfered by some radiant phenomena — like the sound of a busy highway, the light from the sun, or the spray particles emitted from a radioactive source. Current and intensity are now different quantities with different units and different uses, which is why (of course) they use identical symbols.
current  intensity  



Start of a table
 12,000 A current through the magnets of the LHC at CERN