Some Consequences of Maxwell’s Equations

Chapter
Part of the UNITEXT for Physics book series (UNITEXTPH)

Abstract

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
$$\begin{aligned} \begin{array}{l} B=\overrightarrow{\nabla } \wedge \overrightarrow{A}, \\[3mm] \overrightarrow{E}=-\overrightarrow{\nabla } \phi -\frac{1}{c}\frac{\partial \overrightarrow{A}}{\partial t}. \end{array} \end{aligned}$$

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Università di Roma Tor VergataRomeItaly

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