Special Relativity and Maxwell's Equations

Richard E. Haskell

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Table of Contents

Preface                                                                                                        3

I.  SPACE AND TIME                                                                               4

1. Reference Frames                                                                     4

2. The Galilean Transformation                                                  5

3. Velocity Transformation                                                          7

II.  RELATIVISTIC KINEMATICS                                                          9

4.   The Principle of Relativity                                                    9

5.   The Nature of Light                                                                10

6.   The Nature of Time                                                                10

7.   Time Dilation                                                                           13

8.    Length Contraction                                                                16

9.   The Lorentz Transformation                                                  18

10.  Relativistic Velocity Transformation                                   20

11.  Vector Representation of the Lorentz Transformation      23

III.  RELATIVISTIC DYNAMICS                                                              26

12.  Relativistic Momentum                                                          26

13.  Relativistic Energy                                                                  29

14. Transformation of Momentum and Energy                         33

15. Transformation Law for Force                                                36

IV.  MAXWELL’S EQUATIONS                                                              39

16. Coulomb’s Law                                                                        39

17. The Lorentz Force                                                                    41

18. Maxwell’s Equations                                                               44

19. Discussion                                                                                 49

References                                                                                                   51

Preface

The material in this report was originally written as class notes in 1967 for a course I taught in electric and magnetic fields. The derivation of Maxwell’s equations from special relativity and Coulomb’s law was developed at that time in collaboration with Dr. Carl T. Case who was then at the Air Force Avionics Laboratory at Wright-Patterson Air Force Base. We had served in the Air Force together between 1963 and 1966 and had become intrigued with the possible limitations of Maxwell’s equations based on this derivation. After 1970 I moved on to work in other areas including coherent optics, pattern recognition, microprocessors, and embedded systems. Last year I came across this material when cleaning out my office and decided to reprint it in electronic form and make it available on my web site.

If you are interested in understanding special relativity, then you should read Parts I – III. The derivation of Maxwell’s equations from special relativity and Coulomb’s law is given in Part IV. If you just want to find out why this topic is so intriguing then skip directly to the discussion in Section 19.

R. E. Haskell

July 2003