If  the  field  on  the  side  toward  B  is  destroyed,  or  if  the  con-
ductor is  extended  toward  B  in  any  manner,  the  field  on  the  side
toward  A  will  not  be  affected,  and  we  have  the  electric  field
bounded  by  a  concentrated  charge  q  and  the  (induced)  charge
upon  an  infinite  plane  conducting  surface  distant  d  from  q  and
maintained  at  zero  potential.  If  q  is  positive  the  electric  surface
density  is  negative  at  every  point  of  the  surface,  since  D  there
has  the  direction  AB.  The  magnitude  of  D  =  a  is  given  in
(40).  The  total  charge  upon  the  infinite  plane  is  —  q,  since  all
the  tubes  from  A  terminate  upon  the  plane.
ELECTRIC    FIELDS    AND    CONDENSERS.
77


Since  the  field  about  A  was  unaltered  by  the  introduction  of
the  conducting  sheet  and  the  destruction  of  the  field  on  the  side
toward  B,  the  force  between  the  charged  body  at  A  and  the  plane
conductor  is  the  same  as  the  force  formerly  acting  between  A

and  B.     That  is,

F =  -  q2 1 16-rrcd2  (41)

Since  concentrated  charges  do  not  exist,  we  shall  suppose  the
charges  at  A  and  B  distributed  over  extremely  small  conducting


Fig.  22.
spheres  each  of  radius  a,  so  that  the  field  in  the  region  outside
the  spheres  will  be  practically  the  same  as  that  already  discussed.
From  (34),  I.,  the  potentials  of  the  spheres  A  and  B  are


-        and


i2-  22

the  subscripts  i  and  2  being  applied  to  A  and  B  respectively.
Since  the  spheres  are  very  small,  these  equations  become  ver
approximately


78  ELEMENTS   OF    ELECTROMAGNETIC    THEORY.
