For  suppose  a  charge  q  to  be  placed  at  P'  within  the
conductor  $  supposed  at  potential  zero,  the  external
potential  is  everywhere  zero  —  that  is,  the  induced
electrification  produces  an  external  potential  and  electric
force  everywhere  equal  and  opposite  to  that  produced
by  q.  In  this  case,  since  q  is  within  S,  qs,  the  induced
charge  on  S,  is  —  q.  Proceeding  as  before  with  this
value  of  qs,  we  find  that  the  internal  potential  and
electric  force  due  to  the  induced  electrification  is  that
due  to  a  virtual  charge  —  qffat  P,  and  that  for  the
density  at  any  point  E  in  $  weget  as  before

a«_/s     i  PA.  PA'     1

-V  >     (


The  induced  electrification  produced  by  any  given
external  or  internal  electric  system  can  be  found  by
determining  the  distribution  due  to  each  point  of  the
system  and  superimposing  the  distributions.  Hence  an
electric  system  on  the  other  side  of  the  spherical
surface  is  determined  which  would  produce  on  the
same  side  as  the  given  system  the  same  potential  at
every  point  as  is  produced  by  the  induced  distribution.
This  system  is  made  up  of  elements  which  are  the


INDUCED  DISTRIBUTION  ON  CONDUCTING  SPHERE.  81

electric  images  of  the  elements  of  the  inducing  system,    Electric
of  which  it  is  therefore  said  to  be  the  electric  image.       SphfricTl
Thus  let  the  system  be  a  series  of  charges  qv  q2,  &c.,    Surface.
at  external  points  P^  P2>  &c.,  whose  distances  from  the
centre  of  the  sphere  are  flf  /2,  &c.,  the  charge  on  the

sphere  is  then  —  n^-  •  and  the  density  at  any  point  is


the   summation  being  taken   for   every  point   of  the
inducing  system.
