Electric currents in conductors and Ohm's law

In metals the charge carriers are free electrons; in electrolytes both positive and negative ions carry current.

Without an applied field, free electrons move randomly in all directions and there is no net flow; a steady current needs a steady electric field maintained by a source such as a cell.

Ohm's law: the current through a conductor is proportional to the potential difference across it, V = IR, where R is its resistance (unit ohm, Ω).

Resistance depends on the dimensions of the conductor: R = ρl/A, directly proportional to length and inversely to area of cross-section.

Resistivity ρ depends on the material and the temperature, not on the size or shape; its SI unit is Ω m.

Current density j is current per unit area normal to the flow (unit A m⁻²); Ohm's law in vector form is j = σE, where σ = 1/ρ is the conductivity.

When a wire of fixed volume is stretched to n times its length, its area becomes 1/n times, so its resistance becomes n² times.