Drift velocity and mobility

Random motion, zero net current. In a conductor with no applied field, free electrons dash about at thermal speeds of order $10^5$ m/s, bouncing off vibrating lattice ions every $\tau \sim 10^{-14}$ s (the relaxation time). Because every direction is equally likely, the vector sum of all these velocities averages to zero — no field, no preferred direction, no net current, however energetically the electrons are moving individually.

A field adds a small directed nudge. Switch on an electric field $E$. Between collisions, each electron feels a force $-eE$ (its charge is negative, so the force is opposite to $E$) and picks up a small extra velocity along the field direction. The next collision randomises the electron's velocity again, but a new, equally small nudge accumulates before the collision after that. Averaged over many electrons and many collisions, this produces a slow, steady drift velocity $v_d$, directed opposite to $E$:

where: $e$ is the electron charge magnitude, $E$ is the applied field, $\tau$ is the average time between collisions, and $m$ is the electron mass. Because $\tau$ is so short, $v_d$ typically works out to only $10^{-4}$ to $10^{-5}$ m/s in a metal — glacially slow next to the $10^5$ m/s thermal speed.

Mobility scales drift velocity by the field. Since $v_d$ is proportional to $E$, it is natural to define a material property that strips the field dependence out:

where: $\mu$ is the mobility, with SI unit $\text{m}^2\text{V}^{-1}\text{s}^{-1}$. Mobility depends only on the material (through $\tau$ and the effective mass $m$) and on temperature — not on how strong a field you happen to apply. A material with large $\mu$ needs only a weak field to produce a fast drift; this is exactly why semiconductors, engineered for large carrier mobility, can switch quickly at low voltages.

Show that a stream of drifting electrons of number density $n$, drift speed $v_d$, and charge magnitude $e$, crossing a wire of area $A$, produces exactly the current $I = neAv_d$ and current density $J = nev_d$ measured in the circuit.

Step 1. Consider a conductor of cross-sectional area $A$ with free-electron number density $n$ (electrons per unit volume). Pick any cross-section of the wire and watch how many electrons cross it in a short time $\Delta t$.

Step 2. In that time $\Delta t$, every electron drifts a distance $v_d \Delta t$ along the wire. So any electron that starts within a distance $v_d\Delta t$ of the chosen cross-section will cross it before time $\Delta t$ is up. These electrons fill a cylindrical volume of length $v_d\Delta t$ and cross-sectional area $A$, so the volume is $A v_d \Delta t$.

Step 3. The number of electrons in that volume is the density times the volume, $n \, A v_d \Delta t$, and each carries charge of magnitude $e$. So the total charge crossing the section in time $\Delta t$ is

Step 4. Current is charge per unit time, so dividing through by $\Delta t$ gives

and dividing by area gives the current density,

where: $n$ is free-electron density, $e$ is electron charge magnitude, $A$ is cross-sectional area, and $v_d$ is drift speed. This is the microscopic origin of the current an ammeter reads: an enormous $n$ (of order $10^{28}$–$10^{29}\ \text{m}^{-3}$ in metals) compensates for the tiny $v_d$.

A copper wire has cross-sectional area $A = 1.0\times10^{-6}\ \text{m}^2$ and carries a current $I = 1.6\ \text{A}$, with free-electron density $n = 8.5\times10^{28}\ \text{m}^{-3}$. Find the drift velocity, then use it to find the mobility if the field driving this current is $E = 0.02\ \text{V/m}$.

Step 1. Rearrange $I = neAv_d$ for $v_d$:

Step 2. Evaluate the denominator: $(8.5\times10^{28})(1.6\times10^{-19})(1.0\times10^{-6}) = 1.36\times10^{4}$. So

Step 3. Now use $\mu = v_d/E$ with the given field $E = 0.02\ \text{V/m}$:

The drift velocity is about $1.18\times10^{-4}$ m/s, and the corresponding electron mobility in this copper sample is about $5.9\times10^{-3}\ \text{m}^2\text{V}^{-1}\text{s}^{-1}$ — confirming that even though individual electrons crawl at a fraction of a millimetre per second, the sheer number of them crossing the wire each second is enough to sustain an ordinary household current.

• Confusing drift velocity with an electron's actual speed. Drift velocity is a tiny net average superimposed on much faster random thermal motion — it is not how fast any individual electron travels.
• Thinking current takes as long to "start" as an electron takes to travel the wire. The field is set up throughout the circuit almost instantaneously, so current starts everywhere together, even though the electrons themselves drift slowly.
• Sign and direction slip-ups. Conventional current flows along $E$, but electrons (negative charge) drift opposite to $E$; the formula $I = neAv_d$ uses the magnitude of $v_d$, with direction handled separately.
• Treating mobility as field-dependent. Mobility $\mu = e\tau/m$ is a property of the material and temperature; the field only decides how large the resulting drift velocity is, not the value of $\mu$ itself.

Ask the tutor: Want to see how mobility and drift velocity feed directly into deriving Ohm's law and resistivity from the free-electron model — ask the AI tutor to walk through that derivation step by step.