Principles of Electrodynamics

Melvin Schwartz

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486134673
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

Unlike most textbooks on electromagnetic theory, which treat electricity, magnetism, Coulomb's law and Faraday's law as almost independent subjects within the framework of the theory, this well-written text takes a relativistic point of view in which electric and magnetic fields are really different aspects of the same physical quantity.
Suitable for advanced undergraduates and graduate students, this volume offers a superb exposition of the essential unity of electromagnetism in its natural , relativistic framework while demonstrating the powerful constraint of relativistic invariance. It will be seen that all electromagnetism follows from electrostatics and from the requirement for the simplest laws allowable under the relativistic constraint. By means of these insights, the author hopes to encourage students to think about theories as yet undeveloped and to see this model as useful in other areas of physics.
After an introductory chapter establishing the mathematical background of the subject and a survey of some new mathematical ideas, the author reviews the principles of electrostatics. He then introduces Einstein's special theory of relativity and applies it throughout the rest of the book. Topics treated range from Gauss's theorem, Coulomb's law, the Faraday effect and Fresnel's equations to multiple expansion of the radiation field , interference and diffraction, waveguides and cavities and electric and magnetic susceptibility.
Carefully selected problems at the end of each chapter invite readers to test their grasp of the material. Professor Schwartz received his Ph.D. from Columbia University and has taught physics there and at Stanford University. He is perhaps best known for his experimental research in the field of high-energy physics and was a co-discoverer of the muon-type neutrino in 1962. He shared the 1988 Nobel Prize in Physics with Leon M. Lederman and Jack Steinberger.

1

Mathematical Review and Survey of Some New Mathematical Ideas

It would be delightful if we could start right out doing physics without the need for a mathematical introduction. Unfortunately though, this would make much of our work immeasurably more laborious. Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates. Indeed, the beauty and elegance of the physical laws themselves are only apparent when expressed in the appropriate mathematical framework.

We shall try to cover a fair bit of the mathematics we will need in this introductory chapter. Several subjects are, however, best treated within the context of our physical development and will be covered later. It is assumed that the reader has a working familiarity with elementary calculus, three-dimensional vectors, and the complex number system. All other subjects will be developed as we go along.

1-1VECTORS IN THREE DIMENSIONS; A REVIEW OF ELEMENTARY NOTIONS

We begin by reviewing what we have already learned about three-dimensional vectors. As we remember from our elementary physics, there are a large number of quantities that need three components for their specification. Position is, of course, the simplest of these quantities. Others include velocity and acceleration. Even though we rarely defined what was meant by a vector in mathematically rigorous terms, we were able to develop a certain fluency in dealing with them. For example, we learned to add two vectors by adding their components. That is, if r₁ = (x₁, y₁, z₁,) and r₂ = (x₂, y₂, z₂,) are two vectors, then

If a is a number, then

We also found it convenient to represent a vector by means of an arrow whose magnitude was equal to the vector magnitude and whose direction was the vector direction. Doing this permitted us to add two vectors by placing the “tail” of one at the “head” of the other as in Fig. 1-1. We also learned how to obtain a so-called scalar quantity by carrying out a type of multiplication with two vectors. If r₁ = (x₁, y₁, z₁) and r₂ = (x₂, y₂, z₂) are two vectors, then r₁ · r₂ is defined by the equation

It was also shown that r₁ · r₂ could be obtained by evaluating |r₁| |r₂| cos θ₁₂, where |r₁| and |r₂| are, respectively, the magnitudes of r₁ and r₂ and θ₁₂ is the angle between them. Another so-called vector was obtained by taking the cross product of r₁ and r₂. That is,

We shall have much more to say about the true nature of this beast very shortly. At the moment we just recall that it appears in some respects to be a vector whose magnitude is equal to |r₁| |r₂| sin θ₁₂ and whose direction, at right angles to both r₁ and r₂, is given by a so-called right-hand rule in going from r₁ to r₂. If we look from the head toward the tail of r₁ × r₂, we would see the shortest rotation from r₁ to r₂ to be in the counterclockwise direction.

Fig. 1-1 The addition of two vectors can be accomplished by placing...