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electromagnetic field strength

Idea

The field strength of the electromagnetic field.

Details

Over Minkowski space ${ℝ}^4$:

the electromagnetic potential

$$A=\varphi dt+A_{1}d{x}^1+A_{2}d{x}^2+A_{3}d{x}^3$$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

$$F≔dA=E_{1}dt\wedge d{x}^1+E_{2}dt\wedge d{x}^2+E_{3}dt\wedge d{x}^3+B_{1}d{x}^2\wedge d{x}^3+B_{2}d{x}^3\wedge d{x}^1+B_{3}d{x}^1\wedge d{x}^2$$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

$$E_i=\frac{\partial \varphi }{\partial x^i}$$E_i = \frac{\partial \phi}{\partial x^i}

the electric field strength

and

$$B_1=\frac{\partial A_2}{\partial x^3}-\frac{\partial A_3}{\partial x^2}$$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, $dF=0$

this are the first 2 of 4 Maxwell equations

Related concepts

• electromagnetic potential

• Aharonov-Bohm effect

• Dirac charge quantization