How to Calculate a Voltmeter Reading

A voltmeter does not measure voltage at just one point. It measures the potential difference between its two terminals, so the first step is to identify exactly where those terminals connect. Without a circuit diagram and component values, there is no unique numerical reading; the method below applies to common DC resistor circuits.

What a voltmeter reads

If the meter is connected between points A and B, its signed reading is

$$V_{AB}=V_A-V_B.$$

The meter is connected in parallel with the component or section whose voltage you want. In a circuit diagram, trace each meter lead to its connection point. Two points joined by an ideal wire have the same potential, so a meter connected across an ideal wire reads zero. A meter connected across a source reads the source’s terminal voltage, which may differ from its stated electromotive force if the source has internal resistance and is supplying current.

For a DC circuit, the reading’s sign depends on which terminal is connected to which point. Reversing the leads reverses the sign; many ordinary digital meters display a minus sign. For AC, the displayed value is usually an RMS voltage, depending on the meter and waveform.

Calculate the reading in a resistor circuit

Use this sequence:

1. Mark the two points where the voltmeter is connected.
2. Simplify the circuit using series and parallel resistor rules, if possible.
3. Find the voltage at each marked point, or find the voltage across the branch containing the meter.
4. Subtract the potentials in the meter’s lead order, or use Ohm’s law, $V=IR$, for the measured component.

For two resistors in series across a source, the same current flows through both. The voltage across a resistor is its share of the total voltage, proportional to its resistance. This is the voltage-divider rule.

Worked example: meter across one resistor

A $12\,\text{V}$ ideal source drives a $2\,\text{k}\Omega$ resistor in series with a $4\,\text{k}\Omega$ resistor. An ideal voltmeter is connected across the $4\,\text{k}\Omega$ resistor. What does it read?

The total resistance is

$$R_{\text{total}}=2\,\text{k}\Omega+4\,\text{k}\Omega=6\,\text{k}\Omega.$$

The circuit current is

$$I=\frac{12\,\text{V}}{6\,\text{k}\Omega}=2\,\text{mA}.$$

The voltage across the measured resistor is therefore

$$V=(2\,\text{mA})(4\,\text{k}\Omega)=8\,\text{V}.$$

The meter reads $8\,\text{V}$ when its positive lead is on the higher-potential end of the resistor. Equivalently, the voltage-divider calculation is $12\times 4/(2+4)=8\,\text{V}$.

If the meter is connected across both resistors instead, it reads the full $12\,\text{V}$ in this ideal-source example. If its leads are placed across the $2\,\text{k}\Omega$ resistor, it reads $4\,\text{V}$.

Does the voltmeter affect the circuit?

An ideal voltmeter has infinite resistance and draws no current. It can be treated as measuring the circuit voltage without changing the circuit. A real voltmeter has a large but finite input resistance. When connected across a resistor, it is in parallel with that resistor and can slightly change the circuit’s resistance and voltage.

For example, suppose the meter in the previous circuit has an input resistance of $10\,\text{k}\Omega$. Across the $4\,\text{k}\Omega$ resistor, the combined resistance is

$$R_{\text{parallel}}=\frac{(4)(10)}{4+10}\,\text{k}\Omega\approx2.857\,\text{k}\Omega.$$

This combination is in series with the $2\,\text{k}\Omega$ resistor, so the meter’s reading is

$$V_{\text{meter}}=12\,\text{V}\times\frac{2.857}{2+2.857}\approx7.06\,\text{V}.$$

That is lower than the ideal-meter result of $8\,\text{V}$ because the finite-resistance meter loads the circuit. In many basic textbook problems, the intended assumption is an ideal voltmeter unless its resistance is specified.

Common mistakes

• Using the source voltage automatically: A meter reads the voltage between its own terminals, not necessarily the full source voltage.
• Connecting the meter in series in your calculation: A voltmeter is placed in parallel with the element being measured. An ideal voltmeter in series would have infinite resistance and stop current.
• Ignoring the connection points: Components in parallel share the same voltage only when they connect to the same two nodes.
• Forgetting polarity: The magnitude may be correct while the signed reading is negative if the leads are reversed.
• Including meter resistance when it is not needed: Use a finite input resistance only when the problem specifies it or asks about loading.

To get a particular numerical reading, you need the circuit connections and values, plus the voltmeter’s input resistance if the meter is not assumed ideal.