The Physics
Opus in profectus

Electric Current

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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 = q

and instantaneous current for those with no fear of calculus…

I = 
q  =  dq
t dt

The unit of current is the ampère [A], which is named for the French scientist André-Marie 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


The elementary charge is defined to be exactly…

e = 1.602176634 × 10−19 C

The number of elementary charges in a coulomb would be the reciprocal of this number — a repeating decimal with a period of 778,716 digits. I'll write the first 19 digits, which is the most I can possibly write (since arbitrary fractions of the elementary charge don't exist).

C ≈ 6,241,509,074,460,762,607 e

And then I'll write it again with a more reasonable number of digits so it's easier to read.

C ≈ 6.2415 × 1018 e

A current of one ampere is then the transfer of approximately 6.2415 × 1018 elementary charges per second. For those who like coinceidences, this is about the same as ten micromoles.

current density

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 well-developed 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

The unit of current density is the ampère per square meter, which has no special name.

A  =  A

m2 m2

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 = ρ v

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  

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.

ρ1A1v1 = ρ2A2v2

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

q = nqV

V = Ad = Avt

I =  q  =  nqAvt
t t

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

J = nqv


conduction vs. valence electrons, conductors vs. insulators

Drift motion superimposed on thermal motion


Bridge text.

Line graph

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.

Line graph

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.)

Line graph

thermal velocity of an electron in copper at room temperature (classical approximation)…

vrms = √ 3kT
vrms = √ 3(1.38 × 10−23 J/K)(300 K)
(9.11 × 10−31 kg)
vrms ≈  100 km/s  

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

vfermi = √ 2Efermi
vfermi = √ 2(7.00 eV)(1.60 × 10−19 J/eV)
(9.11 × 10−31 kg)
vfermi ≈  1500 km/s  

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)…

vdrift =  1  ∆v =  1  aτ =  1   F  τ =  1   eE  τ
2 2 2 me 2 me
vdrift =  eVτ
vdrift =  (1.60 × 10−19 C)(12 V)(3 × 10−14 s)
2(10 m)(9.11 × 10−31 kg)
vdrift ≈  3 mm/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.


ions, electrolytes


ions, plasma


The symbol I was chosen to represent the intensity of a current by the 19th century French physicist and mathematician André-Marie 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….
André-Marie Ampère, 1826   André-Marie 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 phenomenon — 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
I =  q  

A =  C

t s
I =  P  


A m2

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