nLab
electromagnetic field strength

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

The field strength of the electromagnetic field.

Details

Over Minkowski space 4\mathbb{R}^{4}:

the electromagnetic potential

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

then field strength is the de Rham differential

FdA=E 1dtdx 1+E 2dtdx 2+E 3dtdx 3+B 1dx 2dx 3+B 2dx 3dx 1+B 3dx 1dx 2 F \coloneqq \mathbf{d}A = E_1 \mathbf{d}t \wedge \mathbf{d}x^1 + E_2 \mathbf{d}t \wedge \mathbf{d}x^2 + E_3 \mathbf{d}t \wedge \mathbf{d}x^3 + B_1 \mathbf{d}x^2 \wedge \mathbf{d}x^3 + B_2 \mathbf{d}x^3 \wedge \mathbf{d}x^1 + B_3 \mathbf{d}x^1 \wedge \mathbf{d}x^2

with

E i=ϕx i E_i = \frac{\partial \phi}{\partial x^i}

the electric field strength

and

B 1=A 2x 3A 3x 2 B_1 = \frac{\partial A_2}{\partial x^3} - \frac{\partial A_3}{\partial x^2}

etc

the magnetic field strength.

The field strength is closed, dF=0\mathbf{d} F = 0

this are the first 2 of 4 Maxwell equations

Created on November 9, 2012 17:34:27 by Urs Schreiber (80.187.201.44)