Internal Resistance, Power and the Wheatstone Bridge

Why does a battery get warm?

Use a torch for a long time, and its battery feels warm. The current flows through the battery too, and the battery has some resistance of its own. We met this internal resistance $r$ in the last lesson. Now we see what it does to a circuit.

Emf and terminal voltage

The emf $E$ of a cell is the energy it gives to each coulomb of charge that passes through it. Part of this energy is lost inside the cell, across $r$. What is left is the terminal voltage $V$, the voltage the outside circuit gets.

• With no current, nothing is lost, and $V = E$. A voltmeter across a cell that drives nothing reads its emf.
• The bigger the current, the bigger the lost volts $Ir$, and the smaller $V$.
• A graph of $V$ against $I$ is a straight line. It meets the $V$ axis at $E$, and its slope is $-r$.

An example. A battery of emf $9\ \text{V}$ and internal resistance $0.5\ \Omega$ drives a $4\ \Omega$ lamp. The current is $I = \dfrac{9\ \text{V}}{4\ \Omega + 0.5\ \Omega} = 2\ \text{A}$. The terminal voltage is $V = 9\ \text{V} - 2\ \text{A} \times 0.5\ \Omega = 8\ \text{V}$. So $1\ \text{V}$ is lost inside the battery.

Cells together

• In series, one after another, the emfs add, and so do the internal resistances: $E = E_1 + E_2 + \ldots$ and $r = r_1 + r_2 + \ldots$
• In parallel, $n$ identical cells side by side keep the emf of one cell, but the internal resistance is shared: $r_{\text{total}} = \dfrac{r}{n}$. They can give a larger current for longer.

Three cells of $1.5\ \text{V}$ and $0.3\ \Omega$ each, in series, make a battery of $4.5\ \text{V}$ and $0.9\ \Omega$.

How much power reaches the load?

The load $R$ takes the power $P = I^2 R$, with $I = \dfrac{E}{R + r}$. So

A very small load lets a large current flow, but most of the energy is lost inside the battery. A very large load takes almost no current. In between there is a best value. The power in the load is the largest when $R = r$:

At that point, half of the energy is lost inside the battery. This matching is used in radios and in loudspeakers, where getting the most power out matters more than wasting some.

You should find that the point is highest when it sits on the dashed line, $R = r$, and that a larger emf raises the whole curve.

The Wheatstone bridge

Four resistors $P$, $Q$, $R$ and $S$ are joined in a diamond. A cell is joined across one pair of corners, and a galvanometer across the other pair. When no current flows through the galvanometer, the bridge is balanced, and

If three of the resistors are known, the fourth can be found. The balance does not depend on the emf of the cell, so a weak cell does not spoil the answer.

An example. $P = 100\ \Omega$, $Q = 300\ \Omega$ and $R = 40\ \Omega$. At balance, $S = \dfrac{QR}{P} = \dfrac{300\ \Omega \times 40\ \Omega}{100\ \Omega} = 120\ \Omega$.

A metre bridge is a Wheatstone bridge made with a uniform wire $1\ \text{m}$ long. A sliding contact splits the wire into lengths $l$ and $100 - l$, in centimetres. Their resistances are in the same ratio as their lengths, so at balance $\dfrac{X}{R} = \dfrac{l}{100 - l}$.

Try it in Python

The program finds the power that a battery of $6\ \text{V}$, with $1.5\ \Omega$ inside it, gives to five different loads.

Practise with new numbers

Remember these

• Terminal voltage: $V = E - Ir$. With no current, $V = E$.
• Cells in series: emfs and internal resistances add. Identical cells in parallel: the same emf, and $\dfrac{r}{n}$.
• The power in the load is the largest when $R = r$: $P_{\text{max}} = \dfrac{E^2}{4r}$.
• A balanced Wheatstone bridge: $\dfrac{P}{Q} = \dfrac{R}{S}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A cell of emf $2\ \text{V}$ gives $1.6\ \text{V}$ at its terminals while $0.8\ \text{A}$ flows. What is its internal resistance?

2. Four cells, each of $1.5\ \text{V}$ and $0.25\ \Omega$, are joined in series. What are the emf and the internal resistance of the battery?

3. A battery has an emf of $6\ \text{V}$ and an internal resistance of $1.5\ \Omega$. Which load takes the most power from it, and how much?

4. A metre bridge balances at $40\ \text{cm}$ with an unknown resistor $X$ in the left gap and $6\ \Omega$ in the right gap. What is $X$?