Electromagnetic Waves, Materials, and Computation with MATLAB®

Electromagnetics of Moving Media 261

From the geometry (Figure 14.4), OC = x′

3

, giving

x x x

3 3 4

ʹ

=

( )

+

( )

cos sin .ψ ψ

(14.56a)

Similarly,

ʹ

= −

( )

+

( )

x x x

4 3 4

sin cos .ψ ψ

(14.56b)

Comparing Equation 14.56 with Equation 14.54c, we obtain

tan .ψ β= i

(14.57)

Note that the angle ψ is complex and its cosine cosψ = γ ≥ 1. Thus, Lorentz transforma-

tion as a rotation, by an angle ψ, is a concept rather than a real angular rotation. Figure

14.4b shows L, L

0

, T, and T

0

. The distance L

0

in the frame ∑′ is observed as L in the frame ∑

at a constant time in ∑. Although L appears larger than L

0

in the gure, because of the

complex nature of ψ,

L L

0

= cos