Terminal Voltage
When you connect a battery to a circuit, the voltage you actually measure across its terminals is almost always slightly less than the voltage stamped on the label. A fresh AA battery is marked 1.5 V, yet connect it to a torch bulb and a voltmeter reads perhaps 1.4 V. This real, measurable voltage is called the terminal voltage (or terminal potential difference), and understanding why it differs from the battery's rated voltage is one of the most practically important ideas in circuit electricity. Every real battery or cell behaves as though it contains two parts: a perfect voltage source called the electromotive force (EMF, symbol ε) and a small resistance inside itself called the internal resistance (symbol r). When current flows through the cell, some voltage is 'used up' driving that current through the internal resistance. The voltage drop across r is not available to the external circuit, so the terminal voltage V is always less than ε whenever current is flowing. This is not a flaw — it is an unavoidable consequence of the physics of real materials. Terminal voltage matters enormously in engineering and everyday life. A car battery's terminal voltage falls sharply when the starter motor draws hundreds of amperes, which is why dashboard lights dim momentarily during starting. Understanding the relationship V = ε − Ir allows engineers to design circuits that account for internal resistance and allows physicists to measure the true EMF of a source without touching it.
- What EMF Actually Means
- Internal Resistance: The Hidden Resistor
- Deriving Terminal Voltage
- The Effect of Load Current
- Measuring EMF and Internal Resistance
What EMF Actually Means
EMF (electromotive force) is not really a force — it is an energy-per-unit-charge quantity measured in volts. It represents the total work done by the cell's chemistry (or other energy source) on each coulomb of charge passing through it. Think of the cell as a pump: EMF is the pressure the pump generates before any energy is lost to friction inside the pump itself.
Internal Resistance: The Hidden Resistor
The electrodes and electrolyte inside every real cell have some electrical resistance. This internal resistance r is in series with the external circuit. When current I flows, Ohm's law gives a voltage drop of Ir across this internal resistance. This energy is dissipated as heat inside the cell — you may notice a battery warming up under heavy load. This internal voltage drop is sometimes called the 'lost volts'.
Deriving Terminal Voltage
Applying Kirchhoff's voltage law around the full loop — internal resistance plus external resistance R — gives: ε = IR + Ir. Rearranging, the voltage across the external terminals is V = IR = ε − Ir. This is the terminal voltage: EMF minus the lost volts. When no current flows (open circuit), I = 0 so V = ε; a voltmeter with very high resistance reads the true EMF.
The Effect of Load Current
As the external resistance R decreases (heavier load, more current drawn), the term Ir increases and terminal voltage falls. At the extreme — a short circuit — V drops to zero (all EMF is lost internally) and current reaches its maximum value I_sc = ε/r. Real cells must never be short-circuited; the internal heating can be dangerous. Rechargeable batteries have very small r (often milliohms) to deliver large currents with minimal voltage drop.
Measuring EMF and Internal Resistance
Plot terminal voltage V on the y-axis against current I on the x-axis for different loads. The graph is a straight line with y-intercept ε (the true EMF, read where I = 0) and gradient −r (the internal resistance). This elegant graphical method lets you find both quantities from a single experiment without ever needing to disconnect the cell.
The Key Equation
- V — Terminal voltage (Volt (V)): The actual potential difference measured across the cell's external terminals when current is flowing.
- \varepsilon — Electromotive force (EMF) (Volt (V)): The total energy supplied per coulomb of charge by the cell; equals terminal voltage only when no current flows.
- I — Current (Ampere (A)): The conventional current flowing through the circuit, including through the cell itself.
- r — Internal resistance (Ohm (Ω)): The resistance of the materials inside the cell (electrodes, electrolyte). Causes energy loss as heat within the cell.
Worked Example
Problem. A battery has an EMF of 9.0 V and an internal resistance of 0.50 Ω. It is connected to an external resistor of 4.0 Ω. Calculate: (a) the current in the circuit, and (b) the terminal voltage of the battery.
Answer. The current is 2.0 A and the terminal voltage is 8.0 V.
Origins · 1 minAlessandro Volta and how this was discovered
Volta invented the voltaic pile in 1800 — the first true electrochemical battery — demonstrating that a steady electric current could be produced by chemical action between two different metals and a moist conductor. His work established the concept of an electromotive force (EMF) as a property of a cell, and the unit of potential difference, the volt, is named in his honour. Volta initially developed the voltaic pile to disprove Luigi Galvani's theory of 'animal electricity'. The scientific rivalry between the two Italians inadvertently gave the world its first practical electric battery.
Related Concepts
- Electromotive Force (EMF) — EMF is the source quantity from which terminal voltage is derived; understanding EMF is essential before studying how terminal voltage differs from it.
- Internal Resistance — Internal resistance is the direct cause of the difference between EMF and terminal voltage; the two concepts are inseparable.
- Ohm's Law — The terminal voltage equation V = ε − Ir relies on Ohm's law applied to the internal resistance; Ohm's law underpins the entire analysis.
- Kirchhoff's Voltage Law — Kirchhoff's voltage law around a loop containing the cell formally derives the terminal voltage equation from first principles.