Definition

Potential difference

(E3.1.1) E3.1 · Electric Potential and Potential Difference →

Δ V = \frac{Δ U}{q _{0}} = - \int_{A}^{B} E \cdot d s

Statement

The work the field does per unit charge, with a minus sign, in going from A to B.

Symbols and units

$Δ V$	potential difference	V = J/C
$Δ U$	change in potential energy	J
$q_{0}$	the charge being moved	C
$d s$	step along the path	m
$E$	electric field along the path	N/C

When it applies

Static fields, where the electric force is conservative.

Derivation

Work done by the electric force on a test charge moving a small step:

d W = F \cdot d s = q_{0} E \cdot d s

For a conservative force, the potential energy change is minus that work:

d U = - q_{0} E \cdot d s

Integrate along the path from A to B:

Δ U = - q_{0} \int_{A}^{B} E \cdot d s

Divide by the test charge to get a quantity belonging to the points, not the charge:

Δ V = \frac{Δ U}{q _{0}} = - \int_{A}^{B} E \cdot d s

Worked examples

The electron volt

1 eV = (1.60 \times 1 0^{- 19} C) (1 V) = 1.60 \times 1 0^{- 19} J

Watch out for

V and U are not the same thing: U belongs to the charge-and-field system, V to the point in space. Only differences matter physically; putting V = 0 somewhere is your choice. Released from rest, a positive charge moves towards lower V and a negative charge towards higher V — both move towards lower potential energy.

Connects to

Uniform field Point charge Definition of E