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Currents overview

Topic 1 of 7

Charge flow and drift

Electric current measures the rate of charge flow through a chosen cross-section. A large current can coexist with a very small drift speed because the conductor contains many charge carriers.

If charge Q crosses a section in time t, the mean current is:

I = Q/t
1 A = 1 C/s   and   1 C = 1 A s

For steady current, this mean also equals the current throughout the interval. If the rate changes, Q/t describes that interval's average. Specify the direction when using signed current or charge transfer. Conventional current follows the direction in which positive charge would move.

The elementary charge e is a positive magnitude. Using the supplied value e = 1.60 × 10-19 C, an electron has charge -e. Electrons drifting one way in a metal give conventional current the other way. A resistor transfers energy while charge continues through the steady circuit; it does not use up the charge.

Count the carriers crossing a plane

Consider a uniform conductor of cross-sectional area A. Let n be its number density of mobile carriers, in m-3, and v their mean drift-speed magnitude. Assume one effective carrier type, uniform density and a uniform mean drift velocity over the chosen interval.

Count charge crossing one section

Electrons already occupy the whole metal. The shaded volume is a counting construction for their mean drift, not a packet moving through an otherwise empty wire. The cylinder and particle symbols are schematic.

A one-millimetre drift length sweeps carriers across a marked cross-sectionA horizontal conductor contains electron symbols throughout, on both sides of a marked counting plane. Electrons drift left. The illustrative cylindrical volume immediately to the right of the plane has cross-sectional area one square millimetre and length one millimetre, equal to drift speed 0.000200 metres per second times five seconds. Its mean-drift construction gives volume one times ten to the minus nine cubic metres. The illustrative electron symbols are not the actual carrier count. Conventional current points right, opposite to the mean electron drift. The perspective cylinder is not a scale drawing of the cross-section or the drift length.Counting planeCurrent Iv Δt = 1.00 mmMean electron driftCross-section A = 1.00 mm2

For n = 8.00 × 1028 m-3, this volume contains 8.00 × 1019 carriers in the model. Their charge magnitude is 12.8 C in 5.00 s, giving I = 2.56 A.

Irregular motion and net drift are different

Varied individual motions coexist with a small mean drift opposite to currentSix illustrative electrons have short arrows in varied directions inside a metal region. These represent irregular instantaneous motions, not a quantitative sample from which the drawn net drift is calculated. A separate longer blue arrow points left and is labelled mean electron drift. A brown arrow points right and is labelled conventional current. These two arrows describe different quantities and do not share a magnitude scale. Random motion alone does not establish a sustained net current.Individual irregular motionsMean electron driftConventional current

The sketch does not compare drift speed with current by arrow length. The circuit's electrical response and the much slower drift of an individual electron are also different processes.

The marked plane is the section through which charge is counted. The swept volume is a schematic counting construction in a conductor that already contains carriers throughout. The separate inset distinguishes irregular microscopic motion from mean electron drift and the opposite conventional current.

In time Δt, the mean drift distance is vΔt. The associated volume is A vΔt, so the number of carriers crossing the section in the drift model is N = nA vΔt. Each carrier has charge magnitude |q|:

Q = N|q| = nA vΔt |q|
I = Q/Δt = nAv|q|

The commonly written form is I = nAvq, with q understood as the carrier-charge magnitude when I and v are magnitudes. To use a negative carrier charge in a signed equation, also define an axis and a signed drift velocity consistently. Do not insert electron charge -e while treating everything else as an unsigned magnitude.

Here n is a count per volume, not a total count N or an amount in moles. A means area, while the unit symbol A means ampere. The dimensions confirm the current unit:

m-3 × m2 × (m/s) × C = C/s = A

Estimate the scale

For a rough metal-wire model, take n of order 1029/m3, area 10-6 m2, charge magnitude 10-19 C and drift speed 10-4 m/s. Their product suggests current of order 1 A. Over several seconds, charge can be of order 10 C while a carrier drifts only of order 1 mm. These are rough model scales; use the supplied values for a precise calculation.

Worked carrier model

Slow drift, appreciable current

Use n = 8.00 × 1028 m-3, A = 1.00 mm2 = 1.00 × 10-6 m2, v = 2.00 × 10-4 m/s and |q| = 1.60 × 10-19 C.

I = (8.00 × 1028)(1.00 × 10-6)
× (2.00 × 10-4)(1.60 × 10-19)
= 2.56 A

Over 5.00 s, check each step of the counting model:

Drift distance = vt = 1.00 × 10-3 m = 1.00 mm
Swept volume = Avt = 1.00 × 10-9 m3
N = nAvt = 8.00 × 1019 carriers
Q = N|q| = 12.8 C
Q/t = 12.8/5.00 = 2.56 A

At fixed current, n and |q|, a second wire with half the area needs twice the drift speed: 4.00 × 10-4 m/s. Holding voltage fixed instead would not automatically hold current fixed.

In a metal, rapid irregular electron motion has a much smaller net drift superimposed on it. Irregular motions in opposite directions do not alone produce a sustained net current. Charges already exist throughout the circuit, and its electrical response propagates separately from the slow drift of any one carrier. A particular electron need not travel from the source to the load before the load responds; there is no single universal signal speed for every circuit.

Optional check A wire carries a fixed current with drift-speed magnitude 2.00 x 10^-4 m/s. A second wire has half the cross-sectional area, with the same current, carrier density and carrier-charge magnitude. What is its drift speed?
A wire carries a fixed current with drift-speed magnitude 2.00 x 10^-4 m/s. A second wire has half the cross-sectional area, with the same current, carrier density and carrier-charge magnitude. What is its drift speed?