Internal Resistance

Overview

This note deepens Current Electricity Fundamentals by isolating the behaviour of real sources.

The main Topic 13 idea is simple: when current flows, not all the source’s supplied energy reaches the external circuit because some is dissipated inside the source.

Related topics:

• Electrical Power and Ratings
• Current Electricity Common Exam Traps

Definition

emf $\mathcal{E}$

The emf of a source is the energy supplied per unit charge by the source.

Unit:

$$\text{V}$$

Terminal Potential Difference $V$

The terminal p.d. is the potential difference across the external terminals of the source.

It is the voltage available to the external circuit.

Internal Resistance $r$

Internal resistance is the resistance inside the source itself.

Unit:

$$\Omega$$

Why It Matters

Internal resistance explains:

• why terminal p.d. falls when current increases
• why batteries and power supplies heat up under load
• why short circuits are dangerous
• why an unused cell reads approximately its emf
• why source efficiency falls as more energy is lost inside the source

Key Representations

Source Model

A practical source is modelled as:

• ideal emf source $\mathcal{E}$
• internal resistance $r$ in series with the external circuit

Lost Volts

When current $I$ flows through the internal resistance, the lost volts are

$$Ir$$

Core Equation

Energy conservation around the circuit gives

$$\mathcal{E}=V+Ir$$

so

$$V=\mathcal{E}-Ir$$

Meaning:

• $\mathcal{E}$ is energy supplied per unit charge
• $Ir$ is energy dissipated per unit charge inside the source
• $V$ is energy delivered per unit charge to the external circuit

Open Circuit

When no current flows,

$$I=0$$

so

$$V=\mathcal{E}$$

That is why a voltmeter across an unused cell reads approximately its emf.

Short Circuit

If the external resistance is very small, current is limited mainly by the internal resistance:

$$I_{\text{sc}}=\frac{\mathcal{E}}{r}$$

This can overheat the source and damage it.

V-against-I Graph

From

$$V=\mathcal{E}-Ir$$

a graph of $V$ against $I$ is a straight line:

• vertical intercept = $\mathcal{E}$
• gradient = $-r$
• horizontal intercept = $\mathcal{E}/r$

Power

Power supplied by the source:

$$P_{\text{supplied}}=I\mathcal{E}$$

Power delivered to the external circuit:

$$P_{\text{load}}=IV$$

Power lost inside the source:

$$P_{\text{lost}}=I^{2}r$$

Efficiency

Source efficiency is useful power output divided by total power supplied:

$$\eta =\frac{IV}{I\mathcal{E}}=\frac{V}{\mathcal{E}}$$

Worked Example

A cell has emf $1.5\text{V}$ and internal resistance $0.50\Omega$. It supplies a current of $0.80\text{A}$.

The terminal p.d. is

$$V=\mathcal{E}-Ir$$ $$V=1.5-(0.80)(0.50)=1.1\text{V}$$

The power lost inside the source is

$$P_{\text{lost}}=I^{2}r=(0.80{)}^2(0.50)=0.32\text{W}$$

Common Exam Traps

emf is not a force

emf is energy per unit charge, measured in volts.

Terminal p.d. is not always emf

For a real source supplying current,

$$V<\mathcal{E}$$

Links

• Current Electricity Fundamentals
• Electrical Power and Ratings
• Current Electricity Common Exam Traps
• DC Circuits