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Electromagnetic Potentials

 

Make the following substitutions for E and B:

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

   
$\displaystyle c^2 \boldsymbol{\nabla}\cdot\boldsymbol{A} +\frac{\partial\phi}{\partial t}$ = 0 (3.11)
$\displaystyle \nabla^2\phi - \frac{1}{c^2} \frac{\partial^2\phi}{\partial t^2}$ = $\displaystyle -\frac{\rho}{\epsilon_0}$ (3.12)
$\displaystyle \nabla^2\boldsymbol{A}
- \frac{1}{c^2} \frac{\partial^2\boldsymbol{A}}{\partial t^2}$ = $\displaystyle -\frac{\boldsymbol{j}}
{\epsilon_0 c^2}$ (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.


next up previous index
Next: A More Complicated Field Up: A brief review of Previous: A brief review of
Zdzislaw Meglicki
2001-02-26