Current, Drift Velocity Relation

When you switch on a bulb, it lights up almost instantly — yet the electrons inside the copper wire are crawling forward slower than a snail (a few tenths of a millimetre per second). This is the paradox of drift velocity vs. signal speed, and it's exactly what this concept explains.

Electric current in a conductor arises from the directed drift of free electrons under an applied electric field, superimposed on their random thermal motion.

Rate of flow of charge through a cross-section of a conductor.

The average velocity with which free electrons move opposite to the electric field direction inside a conductor, superimposed on their random motion.

Step 1: Consider a conductor of cross-sectional area A, with free electron density n and drift velocity v_d​.

Step 2: In time t, electrons travel a distance v_dt. Volume swept = Av_dt.

Step 3: Number of free electrons in this volume = nAv_dt.

Step 4: Total charge crossing the section = Q = nAv_dt ⋅ e.

Step 5: Since I = Q/t: I = neAv_d

Step 6: Dividing by area gives current density: j = \[\frac {I}{A}\] = nev_d

Step 7 (Vector form): Since electrons move opposite to E: \[\vec j\] = −ne\[\vec v_d\]

Step 8 (Mobility link): Substituting v_d = μ_eE: j = neμ_eE ⇒ I = neAμ_eE

Problem: Aluminium and copper wires each carry a current of 10 A. Given nAln_{Al}nAl​ and nCun_{Cu}nCu​, find current density and drift velocity for copper (n=8.4×1028 m−3n = 8.4 \times 10^{28}\ m^{-3}n=8.4×1028m−3).