Electric potential

Prof. Dr. Thorsten Jungmann (last updated 2025-10-20)

The electric potential (German: elektrisches Potenzial) describes the potential energy per unit charge at a particular point in the electric field. It is a scalar quantity that indicates how much work is required to move a unit charge from a reference point to this point. The voltage corresponds to the potential difference between two points.

1 Quantity symbol and unit

The quantity symbol of the electric potential is the Greek lowercase letter $\phi$ (‘phi’). The unit is the volt, named after Alessandro Volta, with the unit symbol $\mathrm{V}$:

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

2 Practical and worked examples

The electric potential is defined as the work $W$ that is required to bring a test charge $q$ from infinity (or from a reference node) to the point under consideration:

$$\phi =\frac{W}{q}$$

The voltage $U_{12}$ results from the potential difference between two points:

$$U_{12}={\phi }_1-{\phi }_2$$

2.1 Electric potential of a point charge

The electric potential at a distance $r$ from a point charge $Q$ is:

$$\phi (r)=\frac{1}{4\pi {\epsilon }_0}\cdot \frac{Q}{r}$$

Example: $Q=1.0\mu \mathrm{C}$, $r=10\mathrm{cm}$

$$\phi =\frac{1}{4\pi \cdot 8.854\cdot 10^{-12}F/m}\cdot \frac{1.0\cdot 10^{-6}\mathrm{C}}{0.10\mathrm{m}}=\underline{89.9\mathrm{kV}}$$

2.2 Electric potential between capacitor plates

For a parallel-plate capacitor with a homogeneous electric field $E$ and plate separation $d$:

$$\phi (d)={\phi }_0-E\cdot d$$

Example: $E=5.0kV/m$, $d=2.0\mathrm{mm}$ from the earthed plate

$$\phi =0\mathrm{V}-5.0\cdot 10^{3}V/m\cdot 2.0\cdot 10^{-3}\mathrm{m}=\underline{-10.0\mathrm{V}}$$

2.3 Ground potential as reference node

In practical electric circuits, ground potential is often chosen as the reference node ($\phi =0\mathrm{V}$).

Example of a voltage divider: $U_{supply}=12.0\mathrm{V}$, $R_1=R_2=1.0\mathrm{k\Omega }$

$${\phi }_{mid}=\frac{R_2}{R_1+R_2}\cdot U_{supply}=\frac{1.0}{2.0}\cdot 12.0\mathrm{V}=\underline{6.0\mathrm{V}}$$

2.4 Work in the electric field

The work needed to transport an electric charge $q$ between two points of different electric potential:

$$W=q\cdot ({\phi }_2-{\phi }_1)=q\cdot U_{12}$$

Example: $q=2.0\mathrm{nC}$, ${\phi }_1=100\mathrm{V}$, ${\phi }_2=150\mathrm{V}$

$$W=2.0\cdot 10^{-9}\mathrm{C}\cdot (150-100)\mathrm{V}=\underline{100\mathrm{nJ}}$$

3 Typical orders of magnitude

• Electron shells in atoms: $\phi \approx 1$ to $100\mathrm{eV}$
• Batteries and rechargeable batteries: $\phi \approx 1.5$ to $12\mathrm{V}$
• Household electronics: $\phi \approx 3.3$ to $24\mathrm{V}$
• Mains voltage: $\phi \approx 230\mathrm{V}$ (RMS value)
• High voltage: $\phi \approx 1$ to $1000\mathrm{kV}$
• Thunderclouds: $\phi \approx 10$ to $100\mathrm{MV}$
• Van de Graaff generator: $\phi \approx 1$ to $10\mathrm{MV}$

4 Example phrasings

• ‘The electric potential at point A is ${\phi }_A=150\mathrm{V}$ with respect to ground.’
• ‘The potential difference between the terminals gives the voltage $U=12.0\mathrm{V}$.’
• ‘Ground potential serves as the reference node with ${\phi }_{earth}=0\mathrm{V}$.’
• ‘At a distance of $1.0\mathrm{m}$ from the charged sphere, there is an electric potential of $500\mathrm{kV}$.’
• ‘The midpoint of the voltage divider is at $\phi =6.0\mathrm{V}$.’

5 Related quantities

• Voltage
• Work or energy

6 Related components

• Capacitor: The electric potential determines the stored energy and the field distribution between the plates
• Insulators: Must reliably withstand potential differences without breakdown