EMF and internal resistance

What you'll learn

• What emf really means, and why it is not quite the same as terminal potential difference.
• Why real cells and power supplies have internal resistance.
• How to use the key equation $V=\epsilon -Ir$.
• How to find emf and internal resistance from a graph of terminal p.d. against current.

Prerequisites: charge, current and potential difference

Before internal resistance makes sense, you need three circuit ideas.

Charge is a property of particles such as electrons. It is measured in coulombs, C.

Current is the rate of flow of charge:

$$I=\frac{\Delta Q}{\Delta t}$$

where $I$ is current in amperes, A, $\Delta Q$ is charge in coulombs, C, and $\Delta t$ is time in seconds, s.

Potential difference, often shortened to p.d., is the energy transferred per unit charge between two points in a circuit:

$$V=\frac{W}{Q}$$

where $V$ is potential difference in volts, V, $W$ is energy transferred in joules, J, and $Q$ is charge in coulombs, C.

Using potential difference to find energy transfer

What is emf?

A cell, battery or power supply is an energy source. It transfers energy from some other form into electrical energy.

For example:

• a chemical cell transfers chemical energy into electrical energy
• a solar cell transfers light energy into electrical energy
• a generator transfers kinetic energy into electrical energy

The electromotive force, usually abbreviated to emf, is not actually a force. It is an energy transfer per unit charge.

The unit of emf is the volt, V, because it is also joules per coulomb.

Calculating emf from energy per charge

Terminal p.d. and the real behaviour of a cell

In an ideal world, the p.d. across a cell’s terminals would always equal its emf.

In a real cell, the cell itself has some resistance. When current flows, some energy is dissipated inside the cell, usually as thermal energy. This means the p.d. available across the external circuit is smaller than the emf.

When no current is drawn from a cell, the terminal p.d. is approximately equal to the emf. This is called the open-circuit voltage.

When current flows, the terminal p.d. falls.

Internal resistance

Internal resistance is the resistance inside a source of emf. It is given the symbol $r$ and measured in ohms, Ω.

A real cell can be modelled as:

• an ideal emf source, $\epsilon$
• in series with a small internal resistor, $r$

The external circuit may have a load resistance $R$.

The diagram shows the standard model and the graph you use to analyse it.

The energy lost per coulomb inside the cell is called the lost volts.

For a current $I$ through internal resistance $r$:

$$lost\; volts=Ir$$

So the emf is split between the external circuit and the internal resistance:

$$\epsilon =V+Ir$$

Rearranging gives the most important equation for this topic:

$$V=\epsilon -Ir$$

Combining internal and external resistance

If the external load has resistance $R$, the terminal p.d. across the load is:

$$V=IR$$

The total resistance in the circuit is the external resistance plus the internal resistance:

$$R_{total}=R+r$$

So the current in the circuit is:

$$I=\frac{\epsilon }{R+r}$$

You can also write:

$$\epsilon =I(R+r)$$

This is just conservation of energy per unit charge: the emf is shared between the external load and the internal resistance.

Calculating current and terminal p.d.

The graph method

A very common A-Level method is to measure the terminal p.d. $V$ for different values of current $I$, then plot a graph of $V$ against $I$.

The equation is:

$$V=\epsilon -Ir$$

This has the same form as a straight-line graph:

$$y=c+mx$$

Comparing the two:

• vertical axis: $V$
• horizontal axis: $I$
• y-intercept: $\epsilon$
• gradient: $-r$

So the internal resistance is the negative of the gradient.

Finding emf and internal resistance from a graph

Practical measurement of emf and internal resistance

To investigate a cell experimentally, you can connect:

• an ammeter in series to measure current
• a variable resistor or resistance box to change current
• a voltmeter across the cell terminals to measure terminal p.d.

For each setting of the variable resistor, record $I$ and $V$. Then plot $V$ on the y-axis against $I$ on the x-axis.

Good practical technique matters here:

• switch off between readings to reduce heating
• avoid very high currents, which can damage the cell and change its internal resistance
• use a best-fit line rather than joining points dot-to-dot
• calculate the gradient using a large triangle on the best-fit line
• do not force the graph through the origin

What internal resistance means physically

Internal resistance is not usually a neat little resistor hidden inside the cell. It is a model for several effects inside the source, including energy dissipation as charges move through the electrolyte, electrodes or internal connections.

As current increases, more energy is dissipated per second inside the source. The source may warm up, and the terminal p.d. may fall further.

This also explains why old batteries often perform badly: their internal resistance has increased. Even if the emf is still fairly close to its normal value when measured with a high-resistance voltmeter, the terminal p.d. can drop sharply when the battery is asked to supply a large current.