tailieunhanh - Basic Theoretical Physics: A Concise Overview P18

Basic Theoretical Physics: A Concise Overview P18. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 170 20 Maxwell s Equations II Electromagnetic Waves Hertz s Oscillating Dipole Electric Dipole Radiation Mobile Phones It can be readily shown that the electromagnetic field generated by a timedependent dipole p t at r 0 viz B r t 0curlP 4n r E r t - curl curlP C for t e to to solves everywhere all four Maxwell equations. If B r t obeys Maxwell s equations then so does E r t and vice versa in accordance with . A practical example of such a time-dependent electric dipole so-called Hertz dipole is a radio or mobile-phone antenna driven by an alternating current of frequency w. Since the current is given explicitly by 1 t p t the retarded vector potential has the following form which is explicitly used in equation A r t -p t - . 4nr c In particular the asymptotic behavior of the fields in both the near-field and far-field range can be derived without difficulty as follows a In the near-field range for r C A where A is the wavelength of light in a vacuum corresponding to the frequency w . A - n. one can approximate P 2 by P 2 . After some elementary transformation using the identity curl curlv grad div V2 v and the spherical wave equation given previously one then obtains 3 p t r r r2p t . . E r t 4ngor5 which is the quasi-static result E r t k r 3 for r 0 with which we are already familiar while simultaneously B r t p t x r . k r-2 for r 0 4nr3 and is thus less strongly divergent . asymptotically negligible . E r t for r 0. Here the position-dependence of the denominator dominates. Magnetic Dipole Radiation Synchrotron Radiation 171 b In the far-field range for r A the position-dependence of the numer- ator dominates . Vx p t - r 1 r x -4 c c r Here for r A one obtains asymptotically . with r r r . i r e r t ---z r x r x p t v 7 4 0c2r c B r t - r x p t - 4ncr c and sin 0 2 16n2e0c3r2 p t 012 S E x H r where 9 is the angle between p and r. As for planar waves the propagation vector r r r and the vectors