How to Measure the EMF of a Cell

A cell’s electromotive force (EMF) is the energy it supplies per unit charge. Despite the word “force,” EMF is a voltage, measured in volts. To measure it, you can use a voltmeter for a quick reading or a potentiometer for a null-balance measurement that draws essentially no current from the cell.

The distinction matters because a cell’s voltage at its terminals can change when it supplies current. A loaded-cell reading is not always the cell’s EMF.

Quick method: use a voltmeter

Set a digital multimeter to measure DC voltage, select a range above the cell’s expected voltage, and connect the meter in parallel across the cell: red lead to the positive terminal and black lead to the negative terminal. Read the displayed voltage. Do not connect the meter in current mode across the terminals; that can create a short circuit.

A voltmeter has finite input resistance, so it draws a small current. If the cell has EMF $\mathcal{E}$ and internal resistance $r$, a simple model for a discharging cell gives its terminal voltage as

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

where $I$ is the current supplied. With the meter connected, the meter itself is the load. If its resistance is $R_V$, the model gives

$$V = \mathcal{E}\frac{R_V}{r+R_V}.$$

When $R_V$ is much larger than $r$, the meter draws very little current and $V$ is close to $\mathcal{E}$. For many ordinary measurements, this is accurate enough. The exact input resistance depends on the meter and its range.

More accurate method: use a potentiometer

A potentiometer can measure EMF without drawing current from the cell being tested at the balance point. It uses a uniform resistance wire carrying a steady current from a separate power supply. Because the wire is uniform, the potential drop along it is proportional to its length.

A typical arrangement works as follows:

1. Connect the ends of a uniform wire to a separate driver supply, usually with a rheostat to adjust the current. This establishes a steady potential gradient along the wire.
2. Connect the positive terminal of the test cell to the wire’s high-potential end. Connect the cell’s negative terminal to a galvanometer and a movable contact, called a jockey.
3. Slide the jockey along the wire until the galvanometer reads zero. At this null point, no current flows from the test cell through the measuring circuit.
4. Measure the length $l$ from the high-potential end to the balance point. If the potential drop per unit length is $k$, then

$$\mathcal{E} = kl.$$

To find $k$, first balance a reference cell with known EMF $\mathcal{E}_{\mathrm{ref}}$ at length $l_{\mathrm{ref}}$. Then $k=\mathcal{E}_{\mathrm{ref}}/l_{\mathrm{ref}}$. Keep the driver supply and wire current unchanged when measuring the unknown cell.

Worked example

A $1.50\,\text{V}$ reference cell balances at $75.0\,\text{cm}$. The potential gradient is

$$k=\frac{1.50\,\text{V}}{75.0\,\text{cm}}=0.0200\,\text{V/cm}.$$

An unknown cell balances at $62.5\,\text{cm}$ with the same wire current. Its EMF is

$$\mathcal{E}=(0.0200\,\text{V/cm})(62.5\,\text{cm})=1.25\,\text{V}.$$

Alternatively, using the two balance lengths directly gives $\mathcal{E}_{x}=\mathcal{E}_{\mathrm{ref}}(l_x/l_{\mathrm{ref}})=1.25\,\text{V}$.

Common mistakes

• Measuring in current mode: A current meter has very low resistance. Connecting it directly across a cell can cause a large current and damage the meter or cell. Use voltage mode for a direct terminal measurement.
• Calling every terminal reading the EMF: Under load, the terminal voltage differs from the EMF because of internal resistance. Reduce the load current or use a potentiometer if the distinction matters.
• Reversing the leads: Reversed leads produce a negative reading on many digital meters. Swap the leads or interpret the sign according to the chosen polarity.
• Changing the potentiometer current: The potential gradient changes with wire current. Recalibrate or keep the current fixed when comparing balance lengths.
• Letting the wire heat up: Excessive current can heat the wire, changing its resistance and spoiling the assumption of a stable, uniform potential gradient. Use a modest current and take readings promptly.

For a quick check of a battery, a suitable digital voltmeter is usually sufficient. For a laboratory measurement where loading error must be minimized, the potentiometer’s null method is the better choice.