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Mathematics for Physics

86 Scalar Potential

A scalar field that is used to express the electric field by differentiation, alone in electrostatics and together with the vector potential in electrodynamics.

The line-integral definition of \(V\). The electrostatic potential at a point is minus the line integral of the electric field from a chosen reference point. This principle is used to assign a number \(V\) that depends only on the field point.

The electrostatic potential is

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

where

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

The static gradient relation. In the static case the electric field is minus the gradient of the scalar potential. This principle is used to recover \(\mathbf{E}\) from \(V\).

The static field from the potential is

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

where

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

The electrodynamic reconstruction of \(\mathbf{E}\). In the time-dependent case the electric field also includes minus the time derivative of the vector potential. This principle is used to reconstruct \(\mathbf{E}\) from both potentials.

The electrodynamic field from the potentials is

\[ \mathbf{E} = -\nabla V - \dfrac{\partial\mathbf{A}}{\partial t} \]

where

  • \(\mathbf{E}\) is the electric field.
  • \(V\) is the electric scalar potential.
  • \(\mathbf{A}\) is the magnetic vector potential.
  • \(t\) is time.

The four-potential. The scalar potential is the time part of the electromagnetic four-potential. This principle is used to write \(V\) and \(\mathbf{A}\) as one spacetime vector.

The four-potential is

\[ A^{\alpha} = \Bigl(\dfrac{V}{c},\,\mathbf{A}\Bigr) \]

where

  • \(A^{\alpha}\) is the four-potential.
  • \(V\) is the electric scalar potential.
  • \(\mathbf{A}\) is the magnetic vector potential.
  • \(c\) is the speed of light.

Note: Also called the electric potential. Also denoted \(\phi\).

86.1 References

  1. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(V(\mathbf{r})=-\int_{O}^{\mathbf{r}}\mathbf{E}\cdot d\mathbf{l}\); \(\mathbf{E}=-\nabla V-\dfrac{\partial\mathbf{A}}{\partial t}\).
  2. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — \(A^{\alpha}=(V/c,\mathbf{A})\).
  3. Susskind, L., & Friedman, A. Special Relativity and Classical Field Theory: The Theoretical Minimum. Basic Books, 2017. — gauge scalar freedom of the potentials.