nLab electromagnetic field strength

Idea

The field strength of the electromagnetic field.

Details

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

the electromagnetic potential

$$A = \phi \mathbf{d}t + A_1 \mathbf{d}x^1 + A_2 \mathbf{d}x^2 + A_3 \mathbf{d}x^3$$

then field strength is the de Rham differential

$$F \coloneqq \mathbf{d}A = E_1 \mathbf{d}t \wedge \mathbf{d}x^1 + E_2 \mathbf{d}t \wedge \mathbf{d}x^2 + E_3 \mathbf{d}t \wedge \mathbf{d}x^3 + B_1 \mathbf{d}x^2 \wedge \mathbf{d}x^3 + B_2 \mathbf{d}x^3 \wedge \mathbf{d}x^1 + B_3 \mathbf{d}x^1 \wedge \mathbf{d}x^2$$

with

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

the electric field strength

and

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

etc

the magnetic field strength.

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

this are the first 2 of 4 Maxwell equations

Related concepts

• electromagnetic potential

• Aharonov-Bohm effect

• Dirac charge quantization