Question

According to classical electromagnetic theory, an accelerating electron radiates energy at a rate
Κερα?
  where a is the acceleration, e is the electronic charge, c is the velocity of light, and K is a constant with value of 6 X 10⁹ N m² C^-2. Suppose that the motion of the electron can be represented by the expression during one cycle of its motion.

(a) Show that the energy radiated during one cycle is .

(b) Recalling that the total energy of a harmonic oscillator is where m is the mass, show the the quality factor Q is .

(c) Using a typical value of $\omega$ for a visible photon, estimate the 'lifetime' of the radiating system (e = 1.6 X 10^-19C, mass of an electron = 9.1 X 10^-31kg).

Answer 1

3 Energy radiation rate = de at as accellaration Given a = A sin wt a dx = Aw coswt - la = de dt 1 2 = -Aw² sin wt dt Energy radiation rate = d t = Ke² - A co sin eks" = Kerk we sinust Energy radčated during one cycle, is w=2 you hear wh sin wt dt

2013 Ke skenal sin' (wt).dt 22 2176 73 ll- cos Dcat]) at = keca sina = * = 60,2m) 211/w GE Elf Hell In 1 Jan - . 29/w Il cos zat dt = I sin zato 2, 24 .: Enersy radiated during one cycle - Keaw A = keto? Total envisy b) Quality factor = at Enersy radiated per cycle I. mwhA² 211.c3 - Ke ²ARW mer kew

Energy of oscillator as it damps is, Let = E= I to ( time to decay to half of its initial energy 4 to - top volat =) t--enabled > t = (112) Let wo = 10 15 Hz Q = mc² ке, ense = 1.6x101 c me = 9.1x101 kg 0.693 X 9.1x10 3 2 2 27X1024 $x100x1.5 x1038 x 105 x 1015 t= = ln (2) Q wo = 7.39 x 10 m 170-2 X10-7 sec 28.08 x lotels = 0.73 nsec.