﻿124 
  R. 
  S. 
  Dean 
  — 
  Electron 
  Theory 
  of 
  Passivity. 
  

  

  Let 
  us 
  commence 
  by 
  analyzing 
  mathematically 
  the 
  con- 
  

   ditions 
  affecting 
  the 
  surface 
  concentration 
  of 
  electrons. 
  

  

  If 
  a 
  conductor 
  A,B, 
  with 
  difference 
  of 
  potential 
  E 
  

   between 
  A 
  and 
  B, 
  moves 
  in 
  a 
  magnetic 
  field 
  of 
  intensity 
  

   F, 
  its 
  velocity 
  normal 
  to 
  A,B 
  will 
  be 
  given 
  by 
  

  

  where 
  k 
  is 
  a 
  constant 
  and 
  /* 
  the 
  permeability 
  of 
  the 
  

   medium. 
  Now 
  without 
  interfering 
  with 
  our 
  analysis 
  we 
  

   may 
  replace 
  the 
  conductor 
  A,B 
  with 
  a 
  stream 
  of 
  electrons 
  

   with 
  velocity 
  v, 
  in 
  which 
  case 
  V 
  will 
  be 
  given 
  by 
  

  

  Further, 
  the 
  ends 
  of 
  A,B 
  may 
  be 
  joined 
  and 
  all 
  the 
  elec- 
  

   trons 
  removed 
  but 
  one 
  and 
  equation 
  (2) 
  will 
  still 
  hold. 
  

   We 
  have 
  then 
  our 
  ideal 
  case 
  of 
  one 
  electron 
  moving 
  in 
  its 
  

   orbit 
  with 
  velocity 
  v. 
  Its 
  velocity 
  normal 
  to 
  the 
  plane 
  of 
  

   its 
  orbit 
  will 
  be 
  given 
  by 
  (2). 
  The 
  irregular 
  motions 
  of 
  

   electrons 
  within 
  a 
  metal 
  can 
  be 
  resolved 
  into 
  a 
  series 
  

   of 
  harmonic 
  motions 
  to 
  which 
  this 
  mathematical 
  analysis 
  

   can 
  be 
  applied. 
  If 
  now 
  we 
  consider 
  the 
  surface 
  of 
  a 
  con- 
  

   ductor 
  we 
  find 
  that 
  the 
  electrons 
  on 
  reaching 
  the 
  second 
  

   medium, 
  say 
  air, 
  would 
  have 
  a 
  velocity 
  of 
  

  

  V 
  = 
  V-^, 
  (3) 
  

  

  where 
  V 
  is 
  the 
  velocity 
  in 
  the 
  conductor 
  and 
  f^ 
  the 
  perme- 
  

   ability 
  of 
  the 
  conductor 
  and 
  /*' 
  the 
  permeability 
  of 
  the 
  

  

  second 
  medium. 
  Electrons 
  whose 
  velocity 
  in 
  the 
  con- 
  

  

  y 
  „# 
  

   ductor 
  was 
  greater 
  than 
  -^-, 
  where 
  Ve 
  is 
  the 
  velocity 
  

  

  of 
  escape, 
  will 
  therefore 
  escape. 
  Considering 
  then 
  a 
  

   layer 
  at 
  the 
  interface 
  the 
  number 
  of 
  electrons 
  N 
  coming 
  

   through 
  the 
  lower 
  boundary 
  will 
  be 
  given 
  by 
  

  

  N 
  = 
  d 
  x 
  V 
  (4) 
  

  

  where 
  d 
  1 
  is 
  the 
  electron 
  density 
  and 
  V 
  the 
  velocity. 
  The 
  

   number 
  leaving 
  the 
  upper 
  boundary 
  will 
  be 
  given 
  by 
  

  

  N' 
  = 
  i,V 
  (5) 
  

  

  but 
  N 
  = 
  N' 
  

  

  hence 
  d 
  3 
  = 
  d 
  t 
  , 
  = 
  <l 
  x 
  (6) 
  

  

  V 
  fx 
  

  

  