Real batteries do not behave like perfect energy sources. In AP Physics 2, they are modeled with a simple internal resistor that explains why the voltage available to a circuit falls when current increases.

Modeling a Real Battery

A nonideal battery is treated as two parts combined into one device: an ideal battery, which provides energy to charges, and an internal resistance, which represents energy lost inside the battery itself. In circuit diagrams, this model is equivalent to an ideal source placed in series with a small resistor inside the battery.

Equivalent-circuit model of a real battery: an ideal emf source ($\mathcal{E}$) in series with an internal resistance ($r$). The labeled terminals emphasize that the external circuit sees a reduced voltage when current flows because some potential drop occurs inside the battery across $r$. Source

This simple model captures a major difference between real sources and idealized ones.

When the battery pushes charge through a circuit, the same current must also pass through this internal resistor. That is the key reason a real battery does not always provide its full labeled voltage to the external circuit.

This model does not mean there is literally a single tiny resistor sitting inside the battery. Instead, it is a useful representation of the battery’s chemical and structural limits. For AP Physics 2, the internal resistance is usually treated as a constant value called $r$.

Terminal Potential Difference

The terminal potential difference is the actual potential difference measured across the battery’s external terminals. This is the voltage that the rest of the circuit receives, so it is often smaller than the battery’s emf when current is flowing.

A real battery can therefore be described in two related ways. Its emf is the maximum energy per unit charge the battery can provide ideally, while its terminal potential difference is the part that remains available outside the battery after internal losses are accounted for.

For a battery that is delivering current to a circuit, the internal resistance produces a potential drop of $Ir$ inside the battery.

This equation shows the central idea of the topic: as current increases, the term $Ir$ increases, so the terminal potential difference decreases.

Voltage profile around a simple series circuit showing the battery’s emf rise and the internal potential drop $Ir$ inside the source. The plot makes it clear that the terminal potential difference equals $\mathcal{E}-Ir$, with the remaining drop occurring across the external load. Source

If the battery supplies very little current, the internal voltage drop is small, and the terminal potential difference stays close to the emf.

An important special case occurs when the circuit is open and no current flows. Then $I=0$, so there is no voltage drop across the internal resistance. Under that condition, the terminal potential difference equals the emf. This is why a battery can appear to have its full voltage when tested with almost no load, yet provide less voltage when connected to a device that draws significant current.

How Current Changes the Available Voltage

The amount of voltage “lost” inside the battery depends directly on current.

• Zero current: no internal voltage drop, so $V_{terminal}=\mathcal{E}$.

• Small current: a small internal drop occurs, so terminal potential difference is only slightly less than emf.

• Large current: a larger internal drop occurs, so the terminal potential difference can be much lower than emf.

Because of this, the voltage supplied by a real battery is not fixed by the battery alone. It also depends on how much current the external circuit draws. A device that demands more current causes a greater loss inside the battery, reducing the voltage available at the terminals.

This explains why batteries in high-current situations may seem weaker than expected. The battery is still providing energy, but a larger fraction of that energy is being used up inside the battery rather than being delivered to the external circuit.

Interpreting the Series Model

Thinking of the battery as an ideal source in series with an internal resistor helps with both circuit reasoning and physical interpretation. Since the elements are in series, the current through the ideal source and the internal resistance is the same. The ideal source raises electric potential, while the internal resistance lowers part of it before charge reaches the external circuit.

The model also explains why terminal voltage is sometimes called the loaded voltage of the battery. When a load is attached and current flows, the battery’s internal resistance matters. A larger internal resistance causes a larger internal drop for the same current, so two batteries with the same emf can produce different terminal potential differences in the same situation.

In AP Physics 2 problems, the most important reasoning is usually qualitative:

• A larger current means a smaller terminal potential difference for a discharging battery.

• A larger internal resistance means a smaller terminal potential difference at the same current.

• A battery with no current has terminal potential difference equal to its emf.

These ideas connect the battery’s internal properties to what the external circuit actually experiences. The terminal potential difference is the useful voltage available to the rest of the circuit, while the difference between emf and terminal potential difference represents the energy per unit charge lost inside the battery because of internal resistance.