Electric potential

Characteristics and key properties

• Scalar quantity: potential has magnitude only and no direction, unlike the electric field.
• Relative value: potential is defined only up to an additive constant; only differences in potential (voltage) have direct physical meaning.
• Equipotential surfaces: surfaces on which the potential is constant; no work is done moving a charge along an equipotential.

Mathematical relations

The electric field E and the potential V are related by E = -∇V in regions where the field is well defined; this means the field points in the direction of greatest decrease of V. The potential difference between two points a and b equals the negative line integral of the electric field: ΔV = V(b) - V(a) = -∫_a^b E · dl. For a point charge q located at the origin, the potential at distance r in vacuum is V(r) = k q / r, where k = 1/(4πε_0). For extended charge distributions the potential is obtained by integrating contributions from each charge element.

Calculations and reference choices

Because potential is relative, one commonly chooses the potential to be zero at infinity for isolated charge systems. In circuits and practical settings the reference is often the earth or chassis, called "ground." A voltmeter measures potential differences, not absolute potentials. The work required to move a charge q between two points is W = q ΔV.

Uses, examples, and importance

• In circuits, voltage sources (batteries) maintain a potential difference that drives current.
• Capacitors store energy in the form of potential difference between plates; stored energy depends on voltage and capacitance.
• Electrostatic problems are often solved by computing potentials, which simplifies finding fields via derivatives.
• Measurement and safety standards reference common voltages (household mains, signal levels) in volts to describe risk and design constraints.

Distinctions and notable facts

Electric potential differs from electric potential energy: potential energy scales with the amount of charge (U = qV), while potential is per unit charge. Potential is also distinct from the electric field, which exerts force; potential is a convenient scalar from which the field can be derived. The freedom to add a constant to V is known as gauge freedom. Understanding potential is central to electrostatics, circuit theory, and many applied technologies.