Internal Resistance in Circuits
Every real battery or power source has a small resistance built into it, caused by the physical materials — chemicals, metal contacts, and conducting plates — that make up the cell. This resistance is called internal resistance, and it behaves exactly like any other resistor in the circuit, except that it sits inside the source itself. Because current must flow through it, energy is dissipated inside the battery, which means not all of the battery's stored energy reaches the external circuit. Understanding internal resistance is essential for predicting how a real battery performs under load. The electromotive force (EMF), denoted by the symbol ε, is the total energy supplied per unit charge by the source — think of it as the battery's 'ideal' voltage. However, once current flows, the internal resistance causes a voltage drop inside the battery. The voltage that actually appears across the battery's terminals — the terminal voltage — is always less than the EMF when current is being drawn. This difference is sometimes called the 'lost volts', and it increases as the current drawn increases, which is why a battery powering a high-current device seems weaker than one sitting idle. A practical consequence is that measuring a battery's terminal voltage with a high-resistance voltmeter (which draws almost no current) gives a reading close to the EMF, whereas connecting a load causes the terminal voltage to drop noticeably. This is why a car battery may read 12.6 V at rest but dip to 10 V when the starter motor — drawing hundreds of amperes — is engaged. Internal resistance explains many everyday observations about how batteries behave under different loads.
- The source as two components in series
- Current through the internal resistance
- Calculating terminal voltage
- Power considerations
- Measuring EMF and internal resistance experimentally
The source as two components in series
Model any real battery as an ideal EMF source (ε) connected in series with a small resistor r — the internal resistance. The external circuit connects to the battery's terminals. When no current flows (open circuit), no voltage is dropped across r, so the terminal voltage equals ε exactly.
Current through the internal resistance
When an external load resistance R is connected, a current I flows through the whole loop: both through R and through the internal resistance r. Using Kirchhoff's voltage law around the loop: ε = IR + Ir. The term Ir is the voltage dropped across the internal resistance — these are the 'lost volts'.
Calculating terminal voltage
The terminal voltage V is the voltage across the external resistance R, which equals the voltage measured at the battery terminals. Rearranging the loop equation: V = ε − Ir. This shows clearly that terminal voltage is always less than EMF by the amount Ir. The larger the current or the larger the internal resistance, the greater the drop.
Power considerations
The total power delivered by the EMF source is P_total = εI. This splits into useful power delivered to the external load (P_external = I²R = VI) and wasted power inside the battery (P_internal = I²r). Efficiency of the source is η = P_external / P_total = R / (R + r). A battery with large internal resistance wastes a significant fraction of its energy as heat inside itself.
Measuring EMF and internal resistance experimentally
By varying the external resistance R and measuring both V (terminal voltage) and I (current), you can plot V against I. Rearranging V = ε − Ir into the straight-line form V = −r·I + ε, the y-intercept gives ε and the (negative) gradient gives r. This graph is a standard A-level practical method for determining internal resistance without dismantling the cell.
The Key Equation
- V — Terminal voltage (Volt (V)): The actual voltage measured across the battery terminals when current is flowing.
- \varepsilon — Electromotive force (EMF) (Volt (V)): The total energy supplied per unit charge by the source; the 'ideal' voltage with no internal losses.
- I — Current (Ampere (A)): The electric current flowing through both the internal resistance and the external circuit.
- r — Internal resistance (Ohm (Ω)): The resistance within the battery itself due to its physical construction.
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, (b) the terminal voltage of the battery, and (c) the power wasted inside the battery.
Answer. Current = 2.0 A; Terminal voltage = 8.0 V; Power wasted internally = 2.0 W
Origins · 1 minGeorg Simon Ohm and how this was discovered
Ohm formulated the fundamental relationship between voltage, current, and resistance (Ohm's Law: V = IR) published in 1827. His work on the resistance of conductors — including recognising that sources themselves possess internal resistance affecting the current they can deliver — laid the mathematical groundwork for all circuit analysis. His treatment of cells as having an internal resistance distinct from the external circuit directly underpins the terminal voltage equation V = ε − Ir. Ohm's work was initially dismissed by German academic authorities as 'a web of naked fancies', and he lost his teaching post over it. Recognition came slowly from Britain and France, and he was finally awarded the Royal Society's Copley Medal in 1841 — 14 years after his landmark publication.
Related Concepts
- Electromotive Force (EMF) — EMF is the source voltage that drives current through the circuit; understanding it is a prerequisite for analysing the effect of internal resistance on terminal voltage.
- Ohm's Law — The terminal voltage equation V = ε − Ir applies Ohm's Law (V = IR) to both the internal and external resistances; mastery of Ohm's Law is essential here.
- Kirchhoff's Voltage Law — The loop equation ε = IR + Ir is a direct application of Kirchhoff's Voltage Law, which states that the sum of potential differences around any closed loop equals zero.
- Power in Circuits — Internal resistance causes power to be wasted as heat inside the battery (P = I²r), making power analysis inseparable from understanding internal resistance.
- Series and Parallel Circuits — Internal resistance acts as a resistor in series with the external load; knowing how series resistance affects total resistance and current is directly relevant.