Electrodynamics, chap11

Electrodynamics, chap11 - Chapter 11: Special Theory of...

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Chapter 11: Special Theory of Relativity ( Ref.: Marion & Heald, “Classical Electromagnetic Radiation,” 3rd ed., Ch. 14) Einstein’s special theory of relativity is based on two postulates: 1. Laws of physics are invariant in form in all Lorentz frames (In relativity, we often call the inertial frame a Lorentz frame .) 2. The speed of light in vacuum has the same value c in all Lorentz frames, independent of the motion of the source. The basics of the theory are covered in Appendix A on an elementary level with an emphasis on the Lorentz transformation and relativistic momentum/energy. Here, we examine relativity in the four-dimensional space of x and t , which provides the framework for us to examine the laws of mechanics and electromagnetism. The contents of the lecture notes depart considerably from Ch.11 of Jackson. Instead, we follow Ch. 14 of Marion and Heald. In the lecture notes, section numbers do not follow Jackson.
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0 Consider two Lorentz frames, and . Frame moves along the common -axis with constant speed relative to frame . Assume that at 0, coordinate axe KK K zv K tt ′′ == The Lorentz Transformation: s of frames and overlap. Postulate 2 leads to the following Lorentz transformation for space and time coordinates. [derived in Appendix A, Eq. (A.15), where the relative motion is assumed to be a () 0 2 2 0 2 2 00 0 1 0 long the -axis.] (1) where 1 is the Lorentz factor for the transformation. v c v c x xx yy zz v t z γ = = =− ≡− Section 1: Definitions and Operation Rules of Tensors of Different Ranks in the 4-Dimensional Space ( , , , ) xyzt ′′′ i Frames K and K coincide at 0. = = x y y x , z z 0 v K K
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11.1 Definitions and Operation Rules of … ( continued ) : In many books, the relative speed between two frames is denoted by and the particle velocity in a given frame is denoted by . This eventually leads to two definitions fo A note about notation v u 2 2 2 2 2 2 1 1 r the same notation : Lorentz factor for the transformation, 1 Jackson (11.17) Lorentz factor of a particle in a given frame, 1 Jackson (11.46) and (11 () v c u c γ ⎡⎤ ≡− ⎢⎥ ⎣⎦ 0 . .51) To avoid confusion with the notation (e.g. when we perform a Lorentz transformation of the Lorentz factor of a particle), we will denote the relative speed between two frames by an v 2 2 0 2 2 2 2 1 0 1 d the particle velocity by throughout this chapter, and thus define 1 [Lorentz factor for the transformation] 1 [Lorentz factor of a particle in a given fra v c v c v me].
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00 0 Define a position vector in the 4-dimensional space of and as ( , , ) ( , ) 10 0 0 01 0 0 and a 4-D matrix as t xyzi c t i c t a i μν γγ β ≡= = Four - Dimension Space Quantities and Operation Rules: x x x 0 4 1 0 0 ,/ then, the Lorentz transformation in (1) can be written 0 0 0 0 or (2) and the vc i xx yy xa x i zz i ict ict μμ ν γβ γ = ⎡⎤ ⎢⎥ = ⎣⎦ == 4 1 inverse Lorentz transformation is: . (3) x νμ μ μ= = 4 -vector spatial vector 11.1 Definitions and Operation Rules of …
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Electrodynamics, chap11 - Chapter 11: Special Theory of...

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