nLab
electromagnetic potential

Contents

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Definition

The electromagnetic field on a spacetime XX is mathematically modeled by a circle bundle with connection \nabla on XX. If the underlying bundle is trivial, or else on local coordinate patches nX\mathbb{R}^n \hookrightarrow X over which it is so, this connection is equivalently a differential 1-form AΩ 1( n)A \in \Omega^1(\mathbb{R}^n).

This is then called the electromagnetic potential of the electromagnetic field (sometimes: “vector potential” or “gauge potential of the electromagnetic field”).

Its de Rham differential

FdA F \coloneqq \mathbf{d}A

is the actual field strength of the electromagnetic field.

On a 4-dimensiona Minkowski spacetime with its canonical coordinates {t,x 1,x 2,x 3}\{t,x^1, x^2, x^3\}, the electromagnetic potential AA is naturally expanded into corredinate components, traditionally written as

A=ϕdt+A 1dx 1+A 2dx 2+A 3dx 3. A = \phi \mathbf{d}t + A_1 \mathbf{d}x^1 + A_2 \mathbf{d}x^2 + A_3 \mathbf{d}x^3 \,.

Here

  • ϕ\phi is the electric potential

  • A=[A 1,A 2,A 3]\vec A = [A_1, A_2, A_3] is the magnetic potential

(for this choice of coordinates).

References

Section 5 of

Last revised on May 15, 2014 at 05:27:20. See the history of this page for a list of all contributions to it.