Electric potential and voltage

From force to energy

Electric fields describe force, but many problems are easier using energy. Electric potential, often called voltage, is electric potential energy per unit charge:

V=rac{U}{q}

Potential is a scalar field. It assigns a number to each point in space. The potential difference between two points tells how much electric potential energy changes per unit charge.

Delta V=rac{Delta U}{q}

Therefore,

$Delta U=qDelta V$

The SI unit of potential is the volt:

$1 V=1 J/C$

Work and potential difference

The work done by the electric field is related to potential energy change:

$W_{field}=-Delta U$

For a charge $q$,

$W_{field}=-qDelta V$

If a positive charge moves naturally in the direction of the electric field, its electric potential energy decreases. Negative charges behave oppositely because $q$ is negative.

Potential of a point charge

Taking zero potential at infinity, the electric potential due to a point charge $Q$ is

V=k_erac{Q}{r}

Unlike electric field, potential can be positive or negative depending on the sign of $Q$. It is also a scalar, so potentials from multiple point charges add algebraically:

V_{net}=sum_i k_erac{q_i}{r_i}

This is often easier than vector addition of electric fields.

Electric field and potential

Electric field points in the direction of decreasing electric potential. In one dimension,

E_x=-rac{dV}{dx}

In three dimensions,

abla V$$ The gradient $ abla V$ points in the direction of greatest increase of potential, so the minus sign means the electric field points downhill in potential. For a uniform electric field in the x-direction, $$Delta V=-EDelta x$$ if displacement is along the field direction. ## Equipotential surfaces An equipotential surface is a surface on which $V$ is constant. Moving a charge along an equipotential surface requires no work by or against the electric field because $$Delta V=0$$ Electric field lines are perpendicular to equipotential surfaces. If there were a component of field along an equipotential, a charge would gain or lose energy moving along it, contradicting constant potential. Conductors in electrostatic equilibrium are equipotential objects. ## Electron-volts At atomic scales, energy is often measured in electron-volts. One electron-volt is the energy gained by a charge of magnitude $e$ moving through a potential difference of 1 volt: $$1 eV=1.602 imes 10^{-19} J$$ For example, an electron accelerated through $100 V$ gains kinetic energy of $100 eV$, ignoring relativistic effects and losses. ## Voltage versus absolute potential Only potential differences are usually physically meaningful. The zero of potential can often be chosen conveniently, just like the zero of gravitational potential energy. For point charges, zero is commonly chosen at infinity. In circuits, a ground point is often defined as $0 V$. ## The big idea Electric potential turns electrostatic problems into energy problems. Voltage is potential energy per unit charge, potentials add as scalars, and electric fields are related to spatial changes in potential by $ec{E}=- abla V$. Understanding potential is essential for capacitors, circuits, conductors, and electromagnetic theory.