Some Consequences of Maxwell’s Equations

The (classical) electromagnetic fields in vacuo that satisfy given boundary conditions can be calculated through Maxwell’s equations. In the Gauss system they read as: $$\begin{aligned} \begin{array}{l} \overrightarrow{\nabla }\cdot \overrightarrow{E} = 4 \pi \rho , \\ \overrightarrow{\nabla }\cdot \overrightarrow{B} = 0, \\ \overrightarrow{\nabla }\wedge \overrightarrow{E} = -\frac{1}{c}\frac{\partial \overrightarrow{B}}{\partial t}, \\ \overrightarrow{\nabla }\wedge \overrightarrow{B} = \frac{1}{c}\frac{\partial \overrightarrow{E}}{\partial t}+\frac{4 \pi }{c}\overrightarrow{j}, \end{array} \end{aligned}$$ where \(\overrightarrow{j}\) and \(\rho \) are current density and charge density. In all, they are 4 functions of space and time. The fields can be computed and measured, however, it amazing that the Maxwell equations succeed in giving us 6 measurable quantities (3 components of \(\overrightarrow{E}\) and 3 of \(\overrightarrow{B}\)) having only 4 quantities in input. This is a most remarkable property of the electromagnetic field. Moreover, we can obtain the same field more easily by working out 4 quantities, namely the scalar potential \(\phi \) and the vector potential \(\overrightarrow{A},\) such that $$\begin{aligned} \begin{array}{l} B=\overrightarrow{\nabla } \wedge \overrightarrow{A}, \\[3mm] \overrightarrow{E}=-\overrightarrow{\nabla } \phi -\frac{1}{c}\frac{\partial \overrightarrow{A}}{\partial t}. \end{array} \end{aligned}$$