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Mathematics for Physics

91 Voltage

A potential difference between two points that is used to measure the work per unit charge needed to move charge from one point to the other.

Voltage as work per unit charge. The voltage between two points is the work per unit charge to carry a charge from one point to the other. This principle is used to assign a single number to a pair of terminals.

The voltage as work per unit charge is

\[ V(b) - V(a) = \dfrac{W}{Q} \]

where

  • \(V(a)\) and \(V(b)\) are the potentials at the two points.
  • \(W\) is the work to move charge \(Q\) from \(a\) to \(b\).
  • \(Q\) is the test charge.

Path independence in electrostatics. In a static field the same voltage is obtained along every path between the two points. This principle is used to treat voltage as a property of the pair of points, not of the path.

The line-integral form of the voltage is

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

where

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

Note: Also called electric potential difference.