Electric current and Ohm's law

Electric current is the rate of flow of charge: I = Q / t. Unit: ampere (A), one coulomb per second. 30 C passing in one minute is 0.5 A.

By convention, current flows the way positive charge would move. In a metal wire it is really electrons moving, in the opposite direction.

Inside a metal, the free electrons already rush about at high speed, but randomly, so on average they go nowhere. A battery sets up an electric field in the wire, which adds a small steady drift to this random motion.

The current is I = n e A v_d: n = free electrons per m³, e = electron charge, A = cross-section area, v_d = drift speed.

Drift speeds are tiny, about 1 mm per second. A bulb still lights at once because the field travels along the wire almost at the speed of light, and it starts the electrons drifting everywhere in the wire at the same moment.

Ohm's law: at a fixed temperature, the current in a metal conductor is proportional to the voltage across it: V = IR. R is the resistance, in ohms (Ω).

Resistance depends on the wire's shape and material: R = ρl / A.

• Longer wire: more resistance (R ∝ l).
• Thicker wire: less resistance (R ∝ 1/A).
• ρ, the resistivity, depends only on the material and its temperature, not on the shape.

Example: a wire stretched to twice its length keeps the same volume, so its area halves. Then R = ρ(2l)/(A/2) = 4 times the old value.

Ohm's law in field form: j = σE, where j = I/A is the current density and σ = 1/ρ is the conductivity.