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Mathematics for Physics

239 Potential

A measure of influence at a point in space that is used to compare how strongly a force field acts from place to place.

The additive constant. A potential is defined only up to an additive constant. This principle is used to choose a convenient zero, such as infinity or a grounded conductor.

The gradient relation. A conservative force is minus the gradient of the potential. A conservative force is a force whose work around every closed path vanishes. This principle is used to recover \(\mathbf{F}\) from a single scalar function.

The force from a potential is

\[ \mathbf{F} = -\nabla U \]

where

  • \(\mathbf{F}\) is the force.
  • \(\nabla\) is the gradient.
  • \(U\) is the potential energy.

Superposition of scalars. Potentials are scalars and add by ordinary sums. This principle is used to superpose the potentials of many sources.

Laplace’s equation. In a source-free region the electrostatic potential satisfies Laplace’s equation. This principle is used to solve boundary-value problems between conductors.

Laplace’s equation is

\[ \nabla^{2}V = 0 \]

where

  • \(\nabla^{2}\) is the Laplacian.
  • \(V\) is the electric potential.

Work per unit charge. The potential difference between two points is the work per unit charge or mass along a path joining them. This principle is used to compute kinetic-energy changes without following the force vector.

Equilibrium at a minimum. Stable equilibrium occurs at a local minimum of potential energy. This principle is used to locate rest points of a conservative system.

Note: Also called a potential energy function when it tells how much stored energy is available. Also called a scalar potential when a field can be read from it. In quantum mechanics the potential enters as a position-dependent multiplicative operator.

239.1 References

  1. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — additive constant; scalar character; superposition; work per unit charge.
  2. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists. Academic Press, 2013. — force as negative gradient; quantum multiplicative potential.
  3. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — stored interaction energy; source property; equilibrium.
  4. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — superposition; work interpretation.
  5. Jaffe, R. L., & Taylor, W. The Physics of Energy. Cambridge University Press, 2018. — energy stored in potential.
  6. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — potential operator in quantum mechanics.