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A real battery has EMF $\varepsilon$ (energy per unit charge provided by the chemical reaction) and internal resistance $r$ (resistance of the electrolyte and electrodes). Terminal voltage during discharge: $V = \varepsilon - Ir$ (less than EMF). During charging: $V = \varepsilon + Ir$ (greater than EMF). Short-circuit current: $I_{\max} = \varepsilon/r$.

Cells in series: EMFs add ($\varepsilon_{\text{total}} = \sum \varepsilon_i$), internal resistances add. Used when higher voltage is needed. Cells in parallel (identical): EMF unchanged, internal resistance halved. Used when higher current capacity is needed. For $n$ identical cells in series with external resistance $R$: $I = n\varepsilon/(R + nr)$.

Electrical power: $P = VI = I^2R = V^2/R$. Maximum power transfer occurs when external resistance equals internal resistance ($R = r$), giving $P_{\max} = \varepsilon^2/(4r)$ at 50% efficiency. Joule heating: $H = I^2Rt$. In series, the larger resistor heats more ($H \propto R$ for same $I$). In parallel, the smaller resistor heats more ($H \propto 1/R$ for same $V$). This reversal is one of the most commonly tested concepts.