Absorption of Energy by Electrons. 69 



§ 2. The Disturbed Motion due to Radiation. 



The restrictions assumed in this paper are for the most 

 part unnecessary. It is, however, essential that we should 

 be at liberty to treat the radiation as a small disturbing force 

 which gradually takes the electron out of the orbit it would 

 follow under the internal forces alone. That ensured, we 

 can use the theory of varied motion. The displacement 

 due to radiation must be small during the time of describing 

 a free path — small not only by comparison with the length 

 of the path, but also a small fraction of any wave-length 

 dealt with. 



By a free path I mean the average distance moved by the 

 electron before the direction of its velocity is completely 

 altered. The extent of the deviation actually produced by 

 radiation distributed according to (4) can easily be estimated. 



Let the electric intensity 



a cos p£ + b sin p£ (5) 



act upon a particle of mass m and charge e. Then, if r be 



the vector whose components are the coordinates of the 



particle 



dh 

 ra — = ea cos pt + eb sin pt ; 



and on integration 



x — r — (— J = — -^ (cos pt — 1)— - — ^(s'mpt— pt), 

 t\dt/ mp 2 ^ r J mp 2K l 1 J 



Xq and ( -y- I are the initial position and velocity, and the 



time t is for the free path. The left-hand side of the last 

 equation is the deviation due to the force. Write x p for this, 

 and denote average values by a bar : 



— p 2 a2 e 2 tf 



x p 2 =^# 1 (cos^-l) 2 + -f 1 (sin^-p0 2 . 



The average value of the total energy, electric and 



magnetic, due to (5) is ^— (a 2 + b 2 ) in unit volume. If that 



represents the energy in the interval between p and p + dp, 

 then according to (4) 



and of course a 2 =.b 2 . 



