An external resistance R is powered by a battery of emf E.
Since R is connected directly across the battery, the potential difference across the resistor is always equal to E, regardless of the resistance value of the resistor. But a zero resistance battery is mythical beast, like the frictionless incline and the massless pulley.
In practice, all emf sources carry some inherent internal resistance. For example, a chemical cell must push ions through the electrolyte. An electric generator must push the current through the coil windings with thousands of turns. A solar cell must push electrons and holes through the silicon substrate. If the resistance of the electrical path in the emf source is substantial, then a significant potential difference is required to push the current through the emf source itself. In casual speak, we say that part of the emf of the battery E is used to provide the PD required to push current I through the internal resistance r. The remaining PD, available to the external circuit, is called the terminal potential difference Vt. Hence
Plotted as a graph, we get a downward sloping straight line whose steepness corresponds to the internal resistance r. Notice that the terminal PD is equal to the emf only if zero current is drawn from the battery. This happens if the circuit is left open. This is why the emf is also called the open circuit terminal PD. The smaller the external resistance R, the larger the current drawn, the lower the terminal PD.
An alternative expression for Vt is
This expression comes from the potential divider principle. The emf is divided between the internal and external circuit according to the ratio of the internal and external resistances. The smaller the internal resistance r, the smaller the fraction of emf that is “wasted” across it, and the larger the fraction of emf that pushes current through R (terminal PD).
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