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Electromagnetic Field Theory: A Problem Solving Approach Part 54

Electromagnetic Field Theory: A Problem Solving Approach Part 54. Electromagnetic field theory is often the least popular course in the electrical engineering curriculum. Heavy reliance on vector and integral calculus can obscure physical phenomena so that the student becomes bogged down in the mathematics and loses sight of the applications. This book instills problem solving confidence by teaching through the use of a large number of worked problems. To keep the subject exciting, many of these problems are based on physical processes, devices, and models. This text is an introductory treatment on the junior level for a two-semester electrical engineering. | Sinusoidal Time Variations 505 The transformations of 29 are then z vt t vz co Z . .t . V1 - v c0 2 Vl - v c0 2 34 and are known as the Lorentz transformations. Measured lengths and time intervals are different for observers moving at different speeds. If the velocity v is much less than the speed of light 34 reduces to the Galilean transformations lim z z vt t t 35 vic 1 which describe our usual experiences at nonrelativistic speeds. The coordinates perpendicular to the motion are unaffected by the relative velocity between reference frames x x y y 36 Continued development of the theory of relativity is beyond the scope of this text and is worth a course unto itself. Applying the Lorentz transformation to Newton s law and Maxwell s equations yield new results that at first appearance seem contrary to our experiences because we live in a world where most material velocities are much less than c0. However continued experiments on such disparate time and space scales as between atomic physics and astronomies verify the predictions of relativity theory in part spawned by Maxwell s equations. 7-4 7-4- SINUSOIDAL TIME VARIATIONS Frequency and Wavenumber If the current sheet of Section 7-3-3 varies sinusoidally with time as Re X0 the wave solutions require the fields to j d t l c _J ttu t z c vary as e and e1 Hy Z Í Re Re I k I z c 2 . z 0 z 0 1 2 . jw t z c 2 . Rel 0 jw t z c 506 Electrodynamics Fields and Waves Ac a fixed time the fields then also vary sinusoidally with position so that it is convenient to define the wavenumber as 2ir co 2 where A is the fundamental spatial period of the wave. At a fixed position the waveform is also periodic in time with period T 1 2tt T - f o 3 where is the frequency of the source. Using 3 with 2 gives us the familiar frequency-wavelength formula w kc c 4 Throughout the electromagnetic spectrum summarized in Figure 7-7 time varying phenomena differ only in the scaling of time and size. No matter the frequency or wavelength although .