Fundamentals of Electromagnetic Field Theory

In this chapter we go beyond thermo-mechanics and extend continuum theory to electromagnetic fields. The emphasis is on a rational presentation of fundamental principles: What are the foundations of Maxwell’s equations, how can the occurring fields be measured, at least in principle, and how are they linked to each other? Moreover, the question regarding frame indifference of the equations and the transformation properties of the electromagnetic fields will be posed, which had already been answered before in context with the thermo-mechanical fields. This will lead us to the beginnings of relativistic field theories.

1. 1.

   Becker uses the symbols ρ and g for the (true) electric charge density and the (true) current density of free charge carriers. Both symbols have been used in this book before but in a different context. In what follows we will use the symbols q _f and j _f instead. Also note that Becker does not use SI units, which explains the factor \(4 \uppi \) and the speed of light symbol, c, in his equations.

2. 2.

   For didactic reasons, which will become clearer in Sect. 13.4, we denote the open surface by the symbol S and not like in Sect. 3.3 by the generic symbol for surfaces, A.

3. 3.

   Further down we shall see that the magnetic field, H, has much more complicated transformation properties.

4. 4.

   In the older relativistic literature the stringent application of tensor calculus is avoided and the imaginary unit, i²=−1 is used in context with the definition of the time coordinate. This renders it possible to define the 4D-line element in a quasi-Pythagorean way. If we use tensors from the very beginning on we do not need this concept any more.

5. 5.

   The minus sign in the velocity is arbitrary. In fact some textbooks do not follow this convention. However, we do and this guarantees consistency with the assumed direction of the vector b shown in Fig. 8.1, the remarks in context with Eq. (13.10.3) and, finally, with Exercise 13.11.1.

6. 6.

   Note that it is important to distinguish between co- and contra variant components of the space-time vectors and must strictly be observed in the following formulae. Of course there is no difference between co- and contra variant for Cartesian non-space-time quantities, like the velocity V _i or the rotation matrix O _ij . Consequently, the rule of cross-wise summation (cf., the remark after Eq. (2.4.13), which holds mutatis mutandis also in 4D) does not hold in the subsequent formulae: Unfortunately this diminishes their beauty.

7. 7.

   Spherically, because space is considered to behave isotropically.

Authors and Affiliations

1. Institute of Mechanics, Technical University of Berlin, Berlin, Germany

   Wolfgang H. Müller

Corresponding author

Correspondence to Wolfgang H. Müller .

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