Why Electrons Crawl Through a Wire Yet the Light Comes On at Once

Flip a switch and a bulb glows at once, yet the electrons in the wire move more slowly than an ant crawls. The explanation lies in drift velocity — which also explains why wires resist current and why they heat up.

This part covers electric current, drift velocity and mobility, Ohm's law, and resistivity, temperature effects and power.

**Electric current is the rate of flow of charge through a cross-section, $I=\frac{dq}{dt}$, and in a metal it is carried by free electrons that move randomly at high speed but gain a small net drift opposite to an applied field.

Definitions:

- Steady current** — $I=\frac{q}{t}$; instantaneous current — $I=\frac{dq}{dt}$
- Unit — the ampere, $1$ A $=1$ C s${}^{-1}$
- Conventional direction — the way positive charge would move, opposite to electron flow

Worked example. A current of $0.5$ A flows for $2$ minutes:

$$q=It=0.5\times 120=60\text{ C},n=\frac{q}{e}=\frac{60}{1.6\times 10^{-19}}=3.75\times 10^{20}\text{ electrons}$$

An everyday example. Passengers milling about a railway platform have no net movement, but when a train pulls in, everyone drifts slowly towards the doors — like electrons once a field is applied.

The substance. Current is a scalar despite having a direction, because currents at a junction add algebraically, not as vectors.

**In a field $E$, free electrons gain an average drift velocity $v_d=\frac{eE\tau }{m}$ between collisions, giving a current $I=neA{v}_d$; mobility is $\mu =\frac{v_d}{E}$, and collisions with the lattice make the resistivity $\rho =\frac{m}{n{e}^2\tau }$.

Drift velocity.** Each electron accelerates at $\frac{eE}{m}$ for an average relaxation time $\tau$ between collisions, so

$$v_d=\frac{eE\tau }{m}$$

Current. In time $\Delta t$, electrons within a length $v_d\Delta t$ cross the section, giving $I=neA{v}_d$.

Mobility is the drift velocity per unit field, $\mu =\frac{v_d}{E}=\frac{e\tau }{m}$, measured in m${}^2$ V${}^{-1}$ s${}^{-1}$.

Origin of resistivity. Combining $I=neA{v}_d$, $v_d=\frac{eE\tau }{m}$ and $E=\frac{V}{l}$ gives $V=I\frac{ml}{n{e}^2\tau A}$, so

$$\rho =\frac{m}{n{e}^2\tau }$$

Worked example. A copper wire of cross-section $1\times 10^{-6}$ m${}^2$ carries $1.5$ A, with $n=8.5\times 10^{28}$ m${}^{-3}$:

$$v_d=\frac{I}{neA}=\frac{1.5}{8.5\times 10^{28}\times 1.6\times 10^{-19}\times 10^{-6}}\approx 1.1\times 10^{-4}{\text{ m s}}^{-1}$$

An everyday example. A pipe completely full of marbles pushes one out of the far end the instant one is pushed in — the signal is quick even though each marble barely moves.

The substance. A bulb lights at once because the electric field spreads through the circuit almost at the speed of light, not because electrons race from the switch.

**Ohm's law states that the current through a conductor is proportional to the potential difference across it, $V=IR$, at constant physical conditions; ohmic conductors give a straight-line V-I graph, while devices such as diodes give curved or direction-dependent graphs where the law fails.

The law:**

$$V=IR,R=\rho \frac{l}{A}$$

Worked example. A nichrome wire $2$ m long with cross-section $0.5$ mm${}^2$ has $\rho =1.1\times 10^{-6}\Omega$ m:

$$R=\frac{1.1\times 10^{-6}\times 2}{0.5\times 10^{-6}}=4.4\Omega ,I=\frac{22\text{ V}}{4.4\Omega }=5\text{ A}$$

Ohmic and non-ohmic:

- Ohmic — metals at constant temperature; a straight line through the origin
- Non-ohmic — a curved graph, a graph that depends on the sign of $V$, or more than one $V$ for the same $I$

Limitations of Ohm's law:

- $V$ stops being proportional to $I$, as in a filament that heats up
- The relation depends on the direction of $V$, as in a semiconductor diode
- The relation is not unique, as in gallium arsenide, where one current can occur at two voltages

An everyday example. An old filament bulb has a much higher resistance when glowing than when cold, so its V-I graph bends.

The substance. Ohm's law is not a universal law — it is an experimental rule that many, but not all, conductors obey.

**Resistivity is a material property and conductivity is its reciprocal, $\sigma =\frac{1}{\rho }$; for metals, resistivity rises with temperature as ${\rho }_T={\rho }_0[1+\alpha (T-T_0)]$, and the power used in a resistor is $P=VI=I^{2}R=\frac{V^2}{R}$.

Materials by resistivity:

- Metals — low resistivity, rising with temperature
- Semiconductors — intermediate resistivity, falling with temperature

Why metals behave this way.** Hotter ions vibrate more, so electrons collide more often, $\tau$ falls and $\rho$ rises.

Worked example 1. A copper coil has $R=20\Omega$ at $20^{\circ }$C and $\alpha =4\times 10^{-3}{}^{\circ }$C${}^{-1}$. At $70^{\circ }$C:

$$R=20[1+4\times 10^{-3}\times 50]=20\times 1.2=24\Omega$$

Energy and power. $P=VI=I^{2}R=\frac{V^2}{R}$ and $W=Pt$.

Worked example 2. A $2$ kW geyser runs $30$ minutes a day for $30$ days, using $2\times 0.5\times 30=30$ kWh — ₹240 at ₹8 per unit.

An everyday example. Electricity bills count energy in units, where $1$ unit $=1$ kWh $=3.6\times 10^6$ J.

The substance. Power is sent over long distances at high voltage so that the current, and the $I^{2}R$ loss in the wires, stays small.