Skip to main content

Mathematics for Physics

44 Electric Potential

A scalar field that is used to assign to each point the electrostatic potential energy per unit charge, where a scalar field is an assignment of a number to each point in space.

The potential difference as work per unit charge. The potential difference between two points is the work per unit charge to carry a test charge from one point to the other. This principle is used to replace a vector field problem by a single number at each point.

The potential difference is

\[ V(\mathbf{b}) - V(\mathbf{a}) = -\displaystyle\int_{\mathbf{a}}^{\mathbf{b}}\mathbf{E}\cdot d\mathbf{l} \]

where

  • \(V\) is the electric potential.
  • \(\mathbf{E}\) is the electric field.
  • \(d\mathbf{l}\) is a displacement along the path.

The gradient relation. In electrostatics the electric field is minus the gradient of the potential. This principle is used to recover \(\mathbf{E}\) after \(V\) has been found.

The electric field from the potential is

\[ \mathbf{E} = -\nabla V \]

where

  • \(\mathbf{E}\) is the electric field.
  • \(\nabla\) is the gradient.
  • \(V\) is the electric potential.

The Coulomb potential. The potential of a point charge falls as the inverse of distance. This principle is used to write the potential of a localized charge collection by superposition.

The potential of a point charge is

\[ V(\mathbf{r}) = \dfrac{1}{4\pi\epsilon_{0}}\dfrac{q}{r} \]

where

  • \(V\) is the electric potential.
  • \(q\) is the point charge.
  • \(r\) is the distance from the charge.
  • \(\epsilon_{0}\) is the permittivity of free space.

Note: Potential difference is also called voltage.