Electric Potential Energy and Electric Potential

Electric potential energy is the energy a charge has because of its position in an electric field, U = qV, and the electric potential V at a point is the potential energy per unit charge; for a point charge Q, V = kQ ÷ r, and the potential difference between two points is ΔV = W ÷ q, the work done per unit charge in moving between them. In brief: potential is a scalar measured in volts (1 V = 1 J/C). The electron-volt (eV) is the energy gained by an electron moving through 1 V: 1 eV = 1.602 × 10⁻¹⁹ J. Equipotential surfaces are perpendicular to the field lines, and the field points in the direction in which the potential decreases most rapidly, E = −ΔV ÷ Δx.

Batteries, capacitors and particle accelerators involve moving charges through potential differences. This guide defines electric potential energy and potential, shows how they are related to the field, explains equipotentials and the electron-volt, compares gravitational and electric analogies in a table, works through numerical examples and corrects common mistakes.

What is electric potential energy?

When a charge is moved in an electric field by a force, work is done. If a positive charge is pushed toward another positive charge, work has to be done against the repulsion, and the energy is stored as electric potential energy, just as lifting a mass in a gravitational field stores gravitational potential energy. For two point charges q₁ and q₂, a distance r apart, the potential energy (with zero at infinite separation) is U = kq₁q₂ ÷ r. It is positive for like charges (repulsion, energy must be supplied to bring them together) and negative for unlike charges (they are bound). The electric force is conservative: the work done depends only on the endpoints, and not on the path.

What is electric potential?

The electric potential at a point is the potential energy per unit charge: V = U ÷ q. For a point charge Q, V = kQ ÷ r (zero at infinity). Positive charges create positive potentials, negative charges negative. For several charges, the potentials add algebraically (a scalar sum), which is simpler than adding the fields as vectors. The potential difference (voltage) between points A and B is ΔV = V_B − V_A = W ÷ q, so that a charge q moved through ΔV changes its potential energy by qΔV. A charge released in a field moves spontaneously to lower potential energy: positive charges toward lower potential, negative charges toward higher potential. In a uniform field E, the potential difference over a distance d along the field is ΔV = Ed, so E = V ÷ d. In general the field is the negative gradient of the potential: E = −dV/dx.

The electron-volt is a convenient unit of energy for atomic physics: 1 eV = 1.602 × 10⁻¹⁹ J, the kinetic energy gained by an electron (or any particle with charge e) accelerated through a potential difference of 1 V. The energy needed to ionize hydrogen is 13.6 eV, and a typical chemical bond is a few eV.

What are equipotentials?

An equipotential is a surface (or line) on which all points have the same potential. No work is done in moving a charge along an equipotential, so the electric field is always perpendicular to it. For a point charge, the equipotentials are concentric spheres; for a uniform field, they are parallel planes perpendicular to the field. The surface of a conductor in equilibrium is an equipotential, and the field just outside it is perpendicular to the surface. Closely spaced equipotentials mean a strong field, similar to the contour lines on a map, where closely spaced lines mean a steep slope.

Electric and gravitational analogies

Worked example: potential and energy calculations

1. Potential of a point charge. Q = +10 nC. At r = 0.50 m: V = 8.99 × 10⁹ × 1.0 × 10⁻⁸ ÷ 0.50 = 180 V. At r = 0.10 m: V = 899 V.
2. Work in moving a charge. To move a +1.0 nC charge from 0.50 m to 0.10 m from Q, the work done against the field is W = qΔV = 1.0 × 10⁻⁹ × (899 − 180) = 7.2 × 10⁻⁷ J.
3. Accelerated electron. An electron is accelerated from rest through a potential difference of 1,000 V. Its kinetic energy is qΔV = 1.602 × 10⁻¹⁹ × 1,000 = 1.6 × 10⁻¹⁶ J = 1,000 eV. Its speed: v = √(2KE ÷ m) = √(2 × 1.6 × 10⁻¹⁶ ÷ 9.11 × 10⁻³¹) = 1.9 × 10⁷ m/s, about 6 percent of the speed of light (the nonrelativistic formula is adequate).
4. Proton. A proton falling through 1,000 V gains the same energy, 1,000 eV, but because its mass is 1,836 times greater, its speed is √1,836 = 43 times smaller: v = 4.4 × 10⁵ m/s.
5. Uniform field. Plates 5.0 cm apart with a field of 2.0 × 10⁴ V/m: the potential difference is ΔV = Ed = 2.0 × 10⁴ × 0.050 = 1,000 V.
6. Potential of two charges. At the midpoint between +2.0 µC and −2.0 µC, V = k(2.0 × 10⁻⁶)/r − k(2.0 × 10⁻⁶)/r = 0, although the field there is not zero.

How were these concepts developed?

The concept of potential was introduced by Joseph-Louis Lagrange (1773), Pierre-Simon Laplace and Siméon Poisson, and developed by George Green and Carl Friedrich Gauss. The unit volt was named after Alessandro Volta, inventor of the electric battery (1800). The electron-volt came into use in the early 20th century with the study of atomic and particle physics, such as the acceleration of electrons in X-ray tubes and particle accelerators.

Which mistakes are common?

Mistake 1: Confusing potential and potential energy

Potential is energy per unit charge (V). Potential energy is U = qV (J).

Mistake 2: Thinking that zero potential means zero field

The field depends on the slope of the potential, not its value. The midpoint of a dipole has V = 0 but E ≠ 0.

Mistake 3: Getting the sign wrong for negative charges

Negative charges gain kinetic energy moving to higher potential.

Mistake 4: Adding potentials as vectors

Potential is a scalar. Add values with their signs.