Ohm’s law for a complete circuit

The electrostatic field does no work when a charge goes around a closed loop, so a circuit needs a source of energy based on forces of non-electrostatic nature – extraneous forces (chemical in a battery, magnetic in a generator). The work done by the extraneous forces in moving a unit positive charge inside the source is the electromotive force (EMF) of the source: ℰ = Aext/q, with unit volt. From conservation of energy we get Ohm’s law for a complete circuit: I = ℰ/(R + r), where R is the external and r the internal resistance of the source. Hence ℰ = IR + Ir; the voltage across the terminals is U = IR = ℰ – Ir. In an open circuit (I = 0) U = ℰ; with R = 0 (short circuit) the current is greatest, Isc = ℰ/r – very dangerous, the wires and source overheat. The efficiency of a source is η = Auseful/Aext = R/(R + r) = U/ℰ (×100% for percent); the larger R is, the higher the efficiency.