Rca1948FrequencyModu.. - The New Jersey Antique Radio Club

«/„I|J494 FREQUENCY MODULATION, Volume IX Yx = —; V — —;2 2z =x^=y~ and « = i; ,the solutions of the equations of motion, (1), (2), (3), for the x componentof velocity, x, with the above applied field,iswheree\EF (z) e*"o*o (82)mf•/"H/VoF{z)— I cos »,,(« — z/v^) sin -qe^Hr). (83)2nVThe charge entering the cavity per unit area of the beam betweent„ and t + dt will bedq = —\J\dtwhere J is the current density in the beam before entering the cavity.The current density J in the x direction in the beam between z andz + dz at time tisJcfe = _j|—dz . |e||x = E F (z) e^dzv m v(84)J — E F (z) e« (f -«/ l, » ) .The amplitude of the current density entering (75) and (76), namely,J\n a \ is just the time-independent part of the above expression; hence,e J| | IJm = a) (»o) EoF (*) e~Wv°.(85)This completes the calculation of the first-order approximation for thecurrent.Calculation of the Frequency ShiftThe integral in the denominators of (75) and (76) is easy to compute.The result is