Steady Currents and Circuits

Charge That Keeps Moving

A current density $\mathbf{J}$ carries charge, and charge conservation is the continuity equation:

$$\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}$$

For steady currents nothing accumulates anywhere, so $\nabla \cdot \mathbf{J} = 0$, current in equals current out at every junction. That is Kirchhoff's current law, stated as a field equation.

In an ohmic conductor the response is linear: $\mathbf{J} = \sigma \mathbf{E}$, with $\sigma$ the conductivity and $\rho_{res} = 1/\sigma$ the resistivity. Integrating across a uniform wire gives the familiar

$$R = \frac{\rho_{res} L}{A}, \qquad V = IR$$

Resistance is geometry plus material, long and thin resists, short and fat conducts.

A steady current in a closed loop cannot be driven by an electrostatic field, because $\oint \mathbf{E}\cdot d\mathbf{l} = 0$ means no net energy per loop. Something else must do the work: a generator supplying a non-electrostatic field, whose line integral is the electromotive force

$$\mathcal{E} = \oint \mathbf{E}_{motor} \cdot d\mathbf{l}$$

EMF is measured in volts but is not a potential difference, it is work per unit charge delivered by a chemical, mechanical or magnetic agent. A real source has internal resistance $r$, so its terminal voltage is $\mathcal{E} - Ir$, always below the EMF when delivering current.

The energy balance in a circuit is exact: the source supplies $\mathcal{E}I$, of which $I^2r$ heats the source itself and the rest reaches the load. Joule heating $P = I^2R = V^2/R$ is irreversible, the ordered drift energy ends up as random thermal motion.

Common pitfall: treating EMF and terminal voltage as the same number. They coincide only at zero current. Under load the terminal voltage sags by $Ir$, which is why a failing battery reads fine unloaded and collapses the moment it has to deliver.