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Formulas/physics/Current Electricity/Current and Drift Velocity

Current and Drift Velocity

Current in terms of carrier density n, charge e, cross-sectional area A, and drift velocity vd. Fundamental relation connecting microscopic motion to macroscopic current.
Class 11Class 12
Derivation

Setup

Consider a conductor of cross-sectional area AA with nn free electrons per unit volume, each drifting with speed vdv_d.

Charge crossing a section in time dtdt

In time dtdt, all electrons within a cylinder of length vd dtv_d\, dt and cross-section AA cross the reference plane:

Volume of cylinder=A vd dt\text{Volume of cylinder} = A\, v_d\, dt

Number of electrons in this volume:

dN=n⋅A vd dtdN = n \cdot A\, v_d\, dt

Charge carried:

dQ=e dN=neA vd dtdQ = e\, dN = neA\, v_d\, dt

Current

I=dQdt=neAvdI = \frac{dQ}{dt} = neAv_d I=neAvd\boxed{I = neAv_d}

Verification of units

[neAvd]=m−3⋅C⋅m2⋅m s−1=C s−1=A✓[neAv_d] = \text{m}^{-3} \cdot \text{C} \cdot \text{m}^2 \cdot \text{m s}^{-1} = \text{C s}^{-1} = \text{A} \checkmark
Note
For semiconductors, both electrons ($n$, charge $-e$, drift velocity $v_e$ opposite to $\vec{E}$) and holes ($p$, charge $+e$, drift velocity $v_h$ along $\vec{E}$) carry current: $I = (nev_e + pev_h)A$.