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Mathematics for Physics

56 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.

Note: Also called electric potential difference.

definition [d] (Voltage = Potential Difference) From Griffiths: the potential difference between points \(a\) and \(b\) is equal to the work it takes, per unit charge, to carry a particle from \(a\) to \(b\):

  • \(V(b) - V(a) = \dfrac{W}{Q}\) .

where

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

definition [d] (Voltage) From Knight: it is customary to say that the particle moves through a potential difference \(\Delta V = V_{f} - V_{i}\). The potential difference between two points is often called the voltage.

where

  • \(\Delta V\) is the potential difference.
  • \(V_{i}\) and \(V_{f}\) are the initial and final potentials.

56.1 Elementary Example

56.1.1 Simple

Moving charge \(Q = 2\,\mathrm{C}\) through work \(W = 12\,\mathrm{J}\) gives

\[ \Delta V = \dfrac{W}{Q} = 6\,\mathrm{V} \]

where

  • \(\Delta V\) is the voltage between the endpoints.

56.1.2 General

Three paths from \(a\) to \(b\) in a static field give the same \(\Delta V\).

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

where

  • the line integral is path-independent for electrostatic \(\mathbf{E}\).