Why is the electric potential defined only up to a constant?

Only potential differences have physical meaning because they correspond to work done; the zero of potential is a matter of convention, often chosen at infinity or at a grounded conductor.

Is electrostatic energy stored in the charges or in the field?

Both descriptions give the same total energy, but expressing the energy as an integral of the field's square over space gives a consistent, locally defined energy density that proves essential when fields radiate.

Electric Potential and Energy

The electrostatic potential is the work per unit charge needed to bring a charge from a reference point, and it stores the energy of charge configurations.

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Definition

The electric potential at a point is the electrostatic potential energy per unit charge of a test charge placed there, defined up to an additive constant; the electric field is minus its gradient, and the energy of a configuration is the work required to assemble it.

Scope

This topic treats the scalar electrostatic potential, its relation to the field through the gradient, equipotential surfaces, the potential of multipole expansions, and the energy stored in charge distributions and in the field itself. It includes the work-energy interpretation of potential difference (voltage) and energy density of the electrostatic field.

Core questions

How is the potential obtained from the field, and the field from the potential?
How much energy is stored in assembling a charge distribution?
Where does electrostatic energy reside — in the charges or in the field?

Key concepts

scalar potential
potential difference
voltage
equipotential surface
gradient
energy density
multipole expansion

Key theories

Potential as a path-independent line integral: Because the electrostatic field is conservative (curl-free), the work done moving a charge between two points is path-independent and defines a scalar potential, with the field equal to its negative gradient.
Field energy density: The energy of a configuration can be expressed as an integral of the squared field over all space, supporting the interpretation that energy is stored in the field with a definite local density.

Clinical relevance

Potential and voltage concepts are fundamental to electrochemistry, neurophysiology (membrane potentials), electrocardiography, and the operation of all electronic and power systems.

History

The potential function entered physics through Poisson's and Green's early-nineteenth-century work on the theory of electricity, building on Lagrange's potential in gravitation. Volta's invention of the battery gave a practical source of steady potential difference, and the volt was later named in his honour.

Key figures

Siméon Denis Poisson
George Green
Alessandro Volta

Seminal works

jackson1998
griffiths2017

Frequently asked questions

Why is the electric potential defined only up to a constant?: Only potential differences have physical meaning because they correspond to work done; the zero of potential is a matter of convention, often chosen at infinity or at a grounded conductor.
Is electrostatic energy stored in the charges or in the field?: Both descriptions give the same total energy, but expressing the energy as an integral of the field's square over space gives a consistent, locally defined energy density that proves essential when fields radiate.