Voltage

Prof. Dr. Thorsten Jungmann (last updated 2026-05-04)

Electric voltage (German: elektrische Spannung, also Potenzialdifferenz) is also called potential difference. It is the driving force for the transport of the charge carriers that make up the electric current. Voltage is therefore the cause of the electric current.

1 Quantity symbol and unit

The quantity symbol of electric voltage is $U$.
It is given in the unit volt, named after Alessandro Volta, with the unit symbol V.

$$[U]=\mathrm{V}$$

(read: ‘The unit of the voltage $U$ is the volt.’)

The volt is a derived SI unit.

The capital letter $U$ stands for a constant direct voltage or the constant mean value of a time-dependent alternating voltage.

To denote a time-dependent voltage, the lower-case letter $u=u(t)$ is used for the instantaneous value of the voltage. This is used, for example, when charging and discharging capacitors or capacitances, when switching inductors or inductances on and off, and with alternating voltages.

In complex AC circuit analysis, the quantity symbol $\underline{U}$ is used for the voltage as a complex quantity.

2 Practical and worked examples

2.1 Voltage of a torch battery

The voltage across the terminals of a fresh battery is given like this, for example:

$$U=1.5\mathrm{V}$$

2.2 Voltage when discharging an inductor

The formula for the voltage across the inductor when it discharges through a resistor is, for example:

$$u(t)=-U_0\cdot e^{-\frac{R}{L}t}$$

where:

• $u(t)$ the voltage as a function of time
• $U_0$ the initial voltage
• $e$ Euler’s number ($e\approx 2.71828$)
• $t$ the time
• $R$ the resistance
• $L$ the inductance

This formula describes the exponential decay of the voltage while an inductance discharges.

2.3 Mains voltage

The mains voltage at the socket-outlet is written like this:

$$u(t)=\hat{u}\cdot \sin (\omega t)$$

where:

• $u(t)$ the instantaneous voltage as a function of time
• $\hat{u}$ the peak value (the amplitude) of the voltage, here: $\hat{u}=235\mathrm{V}$
• $\omega$ the angular frequency: $\omega =2\cdot \pi \cdot f$, where:
  • $\pi$ the circle constant, $\pi \approx 3.14159$
  • f the frequency, here: 50 Hz
  • here: $\omega =2\pi f\approx 314{\mathrm{s}}^{-1}$
• $t$ the time

This formula describes the sinusoidal waveform of the alternating current in the power grid.

2.4 Voltage drop across a conductor

According to Ohm’s law, the voltage drop $\Delta U$ along a conductor results from the current $I$, the length $l$ and the conductor cross-section $A$:

$$\Delta U=I\cdot R=I\cdot \rho \cdot \frac{l}{A}$$

where:

• $I$ the current through the conductor
• $R$ the conductor resistance
• $\rho$ the resistivity
• $l$ the effective length of the conductor (in AC circuits, take the outgoing and return conductors into account)
• $A$ the cross-section of the conductor

In practice, the voltage drop in electrical installations must not exceed certain limits. DIN VDE 0100-520 recommends limiting the voltage drop from the origin of the installation to the most distant load to:

• max. 3 % for lighting circuits,
• max. 5 % for other loads (e.g. socket-outlets, electric motors).

The reason: too large a voltage drop leads to power losses, heating of the electric lines and malfunctions of devices (e.g. flickering lamps or starting problems with electric motors).

3 Typical orders of magnitude

While the voltage at the output terminals of a sensor, e.g. a microphone, can be a few microvolts ($\mu \mathrm{V}=10^{-6}\mathrm{V}=0.000001\mathrm{V}$) or less, several hundred kilovolts ($\mathrm{kV}=10^3\mathrm{V}=1000\mathrm{V}$) are present on high-voltage lines in the grids of energy suppliers.

4 Example phrasings

• ‘The fan heater is connected to the mains voltage of 230 V.’
• ‘The miniature fuse is designed for a maximum voltage of 24 V.’
• ‘6.6 V drops across the series resistor of the light-emitting diode.’
• ‘The maximum permissible voltage drop across a standard extension lead is 3 % of the mains voltage.’

5 Related quantities

• Together with the current and the resistance, the voltage appears in the formula for Ohm’s law.

$$U=R\cdot I$$

• The electric power is calculated as the product of voltage and current.

$$P=U\cdot I$$

6 Relevant components

Since a (measurable) voltage drops across (virtually) every component, every component is relevant. In this script, the following components are particularly relevant:

• Battery and rechargeable battery
• Conductor
• Resistor
• Switch
• Capacitor
• Inductor
• Relay and contactor
• Multimeter as a voltmeter