Electromagnetic Potentials

 

Make the following substitutions for E and B:

$$-\boldsymbol{\nabla}\phi - \frac{\partial\boldsymbol{A}}{\partial t}$$ E = (3.9)

$$\boldsymbol{\nabla}\times\boldsymbol{A}$$ B = (3.10)

These substitutions satisfy the Faraday law and the no magnetic monopoles law automatically.

The Gauss' law and the Ampère-Maxwell law are satisfied assuming that:

   

$$c^2 \boldsymbol{\nabla}\cdot\boldsymbol{A} +\frac{\partial\phi}{\partial t}$$ = 0 (3.11)

$$\nabla^2\phi - \frac{1}{c^2} \frac{\partial^2\phi}{\partial t^2} -\frac{\rho}{\epsilon_0}$$ = (3.12)

$$\nabla^2\boldsymbol{A} - \frac{1}{c^2} \frac{\partial^2\boldsymbol{A}}{\partial t^2} -\frac{\boldsymbol{j}} {\epsilon_0 c^2}$$ = (3.13)

Equation (3.11) is   called the Lorentz Gauge.

Electro- and Magento-static systems are described by   Poisson equation, i.e., a Laplace equation   with a nonzero right hand side. The right hand sides of equations (3.12) and (3.13) are referred to as sources.