Terminal voltage is the potential difference measured across a real battery's terminals while it operates; when current I flows out of a battery with emf ε and internal resistance r, the terminal voltage is V = ε − Ir, which is less than the emf and equals the voltage delivered to the external circuit.
Terminal voltage is what a voltmeter actually reads when you clip it across a battery in a working circuit. A real battery isn't a perfect voltage source. Model it as an ideal emf ε in series with a small internal resistance r hiding inside the casing. The moment current flows, some potential is "used up" inside the battery itself, so the voltage that makes it out to the terminals is V_terminal = ε − Ir.
Two limits are worth memorizing. With no current flowing (open circuit, or an ideal voltmeter alone across the battery), Ir = 0 and the terminal voltage equals the emf. With heavy current draw, the Ir drop grows and terminal voltage sags. This is why your car's headlights dim when the starter motor kicks in. If the battery is being charged instead of discharged, current is pushed backward through it and the terminal voltage is actually ε + Ir. Either way, the terminal voltage is the voltage the external circuit actually receives, so it's the number you use for the rest of your circuit analysis.
Terminal voltage lives in Topic 11.5 (Compound Direct Current Circuits) in AP Physics C: E&M, where the CED expects you to analyze circuits with nonideal batteries. It's the bridge between the idealized circuits of earlier topics and realistic ones. Conceptually, it's just Kirchhoff's loop rule applied with the battery's internal resistance included as one more series resistor. If you can write ε − Ir − IR_eq = 0, you've already derived everything about terminal voltage. The exam loves this because it tests whether you understand that emf is a property of the battery while terminal voltage depends on the circuit you attach to it.
Keep studying AP® Physics C: E&M Unit 11
Internal Resistance (Topic 11.5)
Internal resistance is the reason terminal voltage exists as a separate idea at all. Treat r as just another series resistor inside the battery, and terminal voltage becomes the potential at the point where the wire leaves the casing.
Nonideal Battery (Topic 11.5)
The nonideal battery model is emf plus internal resistance in series. Terminal voltage is the measurable output of that model, which is why graphing V_terminal vs. I gives a line with y-intercept ε and slope −r, a classic experimental-design setup.
Voltage Divider (Topic 11.5)
A battery with internal resistance driving an external resistance R is literally a voltage divider. The terminal voltage is the fraction R/(R + r) of the emf, which is why a larger external resistance pulls the terminal voltage closer to ε.
Series and Parallel Connections (Topic 11.5)
In compound circuits you first collapse the resistor network to one equivalent R, then add r in series with it to find the current. Get R_eq wrong and your terminal voltage answer is wrong too, so these skills are tested together.
Multiple-choice questions usually hand you ε, r, and an external resistor network, then ask for the terminal voltage. The recipe is always the same. Find the equivalent external resistance, compute I = ε/(R_eq + r), then take V = ε − Ir. For example, a 12.0 V battery with r = 0.50 Ω driving 3.0 Ω and 6.0 Ω in series gives I = 12.0/9.5 ≈ 1.26 A and a terminal voltage of about 11.4 V. Tougher stems flip it into a ratio problem: if V_terminal = 0.75ε, then Ir = 0.25ε, which forces R_eq = 3r, and you reason from there about what happens when R doubles. On FRQs, expect terminal voltage inside experimental-design or circuit-analysis parts. A common task is interpreting a V vs. I graph for a battery, where the intercept is the emf and the magnitude of the slope is the internal resistance. Watch for distractors that add wire resistance; lump it into the external resistance, not into r.
Emf (ε) is the fixed, ideal voltage the battery's chemistry can supply, while terminal voltage is what actually appears across the terminals once current flows. They're only equal when I = 0. During discharge, terminal voltage drops below emf by Ir; during charging, it sits above emf by Ir. If a problem says "a battery labeled 12 V," that's the emf, and the circuit almost never sees the full 12 V.
Terminal voltage during discharge is V = ε − Ir, so it is always less than the emf whenever current flows out of the battery.
With zero current (open circuit), terminal voltage equals the emf, which is how you measure ε with an ideal voltmeter.
The terminal voltage equals the total voltage delivered to the external circuit, so it's the value you use for power and current calculations outside the battery.
To find terminal voltage in a compound circuit, reduce the external network to one equivalent resistance, solve I = ε/(R_eq + r), then compute ε − Ir.
On a graph of terminal voltage versus current, the y-intercept gives the emf and the slope's magnitude gives the internal resistance.
When a battery is being charged, current runs backward through it and the terminal voltage rises to ε + Ir.
It's the actual potential difference across a battery's terminals while current flows, equal to V = ε − Ir where ε is the emf, I is the current, and r is the internal resistance. It's the voltage the external circuit really receives.
No, except in one special case. Terminal voltage equals emf only when no current flows (open circuit). Once current is drawn, the internal resistance eats Ir volts inside the battery, so terminal voltage is less than emf.
Yes, when the battery is being charged. Current is forced backward through the battery, so the loop rule gives V = ε + Ir, putting the terminal voltage above the emf.
Collapse the external resistors to one equivalent resistance R, find the current I = ε/(R + r), then compute V = ε − Ir. For a 12.0 V battery with r = 0.50 Ω driving 9.0 Ω of external resistance, I ≈ 1.26 A and V ≈ 11.4 V.
For a simple series circuit, they're the same number, since the terminal voltage is exactly what gets dropped across the external circuit. The distinction matters in compound circuits, where the terminal voltage splits among multiple resistors according to series and parallel rules.
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