Basic Theoretical Physics A Concise Overview

16 Introduction and MathematicalPreliminaries to Part IIThe theory of electrodynamics provides the foundations for most of ourpresent-day way of life, e.g., electricity, radio and television, computers, radar,mobile phone, etc.; Maxwell’s theory lies at the heart of these technologies,and his equations at the heart of the theory. Exercises relating to this part of the book (originally in German, but nowwith English translations) can be found on the internet, [2]. Several introductory textbooks on theoretical electrodynamics and as-pects of optics can be recommended, in particular “Theoretical Electrody-namics” by Thorsten Fliessbach (taken from a comprehensive series of text-books on theoretical physics). However, this author, like many others, exclu-sively uses the Gaussian (or cgs) system of units (see below). Another bookof lasting value is the text by Bleaney and Bleaney, [10], using mksA units,and containing an appendix on how to convert from one system to the other.Of similar value is also the 3rd edition of the book by Jackson, [11].16.1 Different Systems of Units in ElectromagnetismIn our treatment of electrodynamics we shall adopt the international sys-tem (SI) throughout. SI units are essentially the same as in the older mksAsystem, in that length is measured in metres, mass in kilograms, time inseconds and electric current in amp`eres. However other systems of units,in particular the centimetre-gram-second (cgs) (or Gaussian) system, arealso in common use. Of course, an mks system (without the “A”) wouldbe essentially the same as the cgs system, since 1 m = 100 cm and1 kg = 1000 g. In SI, the fourth base quantity, the unit of current, amp`ere(A), is now defined via the force between two wires carrying a current. Theunit of charge, coulomb (C), is a derived quantity, related to the amp`ereby the identity 1 C = 1 A.s. (The coulomb was originally defined via theamount of charge collected in 1 s by an electrode of a certain electrolytesystem.) For theoretical purposes it might have been more appropriate toadopt the elementary charge |e| = 1.602 . . . · 10−19C as the basic unit ofcharge; however, this choice would be too inconvenient for most practicalpurposes.

110 16 Introduction and Mathematical Preliminaries to Part II The word “electricity” originates from the ancient Greek word ελεκτ oν,meaning “amber”, and refers to the phenomenon of frictional or static elec-tricity. It was well known in ancient times that amber can be given an electriccharge by rubbing it, but it was not until the eighteenth century that the lawof force between two electric charges was derived quantitatively: Coulomb’slaw1 states that two infinitesimally small charged bodies exert a force on eachother (along the line joining them), which is proportional to the product ofthe charges and inversely proportional to the square of the separation. In SICoulomb’s law is written F 1←2 = q1q2 e1←2 , (16.1) 4πε0r12,2where the unit vector e1←2 describes the direction of the line joining thecharges,e1←2 := (r1 − r2)/|r1 − r2| , while r1,2 := |r1 − r2|is the distance between the charges. According to (16.1), ε0, the permittivity of free space, has the physicaldimensions of charge2/(length2 · f orce). In the cgs system, which was in general use before the mksA system hadbeen introduced2, the quantity 4πε0 in Coulomb’s law (16.1) does not appearat all, and one simply writes F 1←2 = q1q2 e1←2 . (16.2) r12,2Hence, the following expression shows how charges in cgs units (primed)appear in equivalent equations in mksA or SI (unprimed) (1a): q ⇔√q , 4πε0i.e., both charges differ only by a factor, which has, however, a physicaldimension.(Also electric currents, dipoles, etc., are transformed in a similar way toelectric charges.) For other electrical quantities appearing below, the relationsare different, e.g., (2a): √ E ⇔ E · 4πε0(i.e., q · E = qE(≡ F )); and, (3a): D ⇔ D · 4π ε01 Charles Augustin de Coulomb, 1785.2 Gauß was in fact an astronomer at the university of G¨ottingen, although perhaps most famous as a mathematician.