nLab
electromagnetic potential

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

Revised on May 15, 2014 05:27:20 by Urs Schreiber (145.116.148.168)