﻿between 
  Cathode 
  and 
  Rontgen 
  Rays. 
  181 
  

  

  this 
  circuit 
  is 
  equal 
  to 
  

  

  47T&H 
  y 
  + 
  iire. 
  

  

  This 
  is 
  equal 
  to 
  4-7T 
  times 
  the 
  change 
  in 
  the 
  electrostatic 
  

   polarization 
  through 
  the 
  same 
  circuit 
  : 
  now 
  when 
  the 
  particle 
  

   was 
  stopped, 
  this 
  polarization 
  was 
  zero, 
  and 
  when 
  the 
  field 
  has 
  

   reached 
  a 
  steady 
  state, 
  the 
  electric 
  intensity 
  is 
  uniformly 
  

   distributed, 
  so 
  that 
  the 
  polarization 
  through 
  the 
  circle 
  is 
  

  

  ^(l-cos*), 
  

  

  where 
  6 
  is 
  the 
  acute 
  angle 
  between 
  OP 
  and 
  the 
  axis 
  of 
  z, 
  P 
  

   being 
  a 
  point 
  on 
  the 
  circumference 
  of 
  the 
  circle 
  ; 
  hence 
  

  

  7 
  

  

  47r6H 
  ^ 
  + 
  kire 
  = 
  Zire 
  (1 
  — 
  cos 
  6), 
  

  

  1 
  Ye 
  cot 
  -r 
  

   or 
  1 
  ^ 
  

  

  H=-± 
  

  

  2 
  rd 
  

  

  where 
  r 
  is 
  the 
  distance 
  OP. 
  The 
  minus 
  sign 
  denoting 
  that 
  

   the 
  magnetic 
  force 
  in 
  the 
  spherical 
  pulse 
  is 
  in 
  the 
  opposite 
  

   direction 
  to 
  that 
  in 
  the 
  plane 
  pulse. 
  

  

  At 
  a 
  point 
  behind 
  the 
  charged 
  particle 
  there 
  is 
  no 
  plane 
  

   pulse, 
  so 
  that 
  

  

  4tt6H' 
  ^= 
  -2w*(l-COS 
  6 
  1 
  ), 
  

  

  where 
  6' 
  is 
  the 
  acute 
  angle 
  between 
  OP 
  and 
  the 
  axis 
  of 
  z 
  ; 
  

  

  thus 
  

  

  j 
  

  

  HT— 
  i 
  

  

  V<?tan£ 
  

  

  2 
  rd 
  

  

  hence 
  if 
  is 
  the 
  angle 
  between 
  OP 
  and 
  the 
  positive 
  direc- 
  

   tion 
  of 
  the 
  axis 
  of 
  z, 
  the 
  magnetic 
  force 
  at 
  any 
  point 
  in 
  the 
  

   spherical 
  wave 
  is 
  given 
  by 
  

  

  Thus 
  we 
  see 
  that 
  the 
  stoppage 
  of 
  a 
  charged 
  particle 
  will 
  

   give 
  rise 
  to 
  very 
  thin 
  pulses 
  of 
  intense 
  magnetic 
  force 
  and 
  

   electric 
  intensity; 
  when 
  the 
  velocity 
  of 
  the 
  particle 
  is 
  small 
  

   there 
  will 
  be 
  one 
  spherical 
  pulse; 
  when 
  the 
  velocity 
  is 
  nearly 
  

   equal 
  to 
  that 
  of 
  light 
  there 
  will 
  in 
  addition 
  to 
  the 
  spherical 
  

   pulse 
  be 
  a 
  plane 
  one 
  propagated 
  only 
  in 
  the 
  direction 
  in 
  

  

  