Theory — Current Density and Drift Velocity
Electric current is the rate of charge flow. In a uniform wire, the current density (vector) is defined as
\( \mathbf{J} = I/A\;\hat{\ell} \),
where \(A\) is the cross-sectional area and \(\hat{\ell}\) is the unit direction of conventional current.
For magnitudes in 1D you often write:
\( J = I/A \).
Microscopic model
In a conductor, many mobile charge carriers (often electrons) drift under an electric field. A common microscopic relation is:
\( \mathbf{J} = n\,q\,\mathbf{v}_d \),
where:
Solving for drift velocity:
\( \mathbf{v}_d = \mathbf{J}/(nq) \).
If the carriers are electrons (\(q=-e\)), then \(\mathbf{v}_d\) points opposite \(\mathbf{J}\) (opposite conventional current).
Why is drift so slow?
Typical metals have huge carrier densities (\(n\sim 10^{28}\)–\(10^{29}\,\mathrm{m^{-3}}\)), so the required drift speed to
carry several amps is often only \(10^{-4}\) m/s (fractions of a mm/s).
This does not mean “electricity is slow”: the electromagnetic signal and energy transfer propagate through the circuit
much faster than individual carrier drift.
Typical carrier densities (order-of-magnitude)
University tease: Hall effect
In the Hall effect, a magnetic field deflects moving carriers and creates a transverse voltage. Its sign reveals whether the
dominant carriers are electrons (\(q<0\)) or holes/positive carriers (\(q>0\)).
The calculator’s animation exaggerates drift so you can see motion—real drift is typically far too slow to notice.
