Dielectric  Strength  of  Air,  243 
easily  find  that 
dv  Yah 
dr      r2  (b  —  a)  ' 
where  V  is  the  P.D.  between  the  spheres.     Now  dv/dr  has 
obviously  its  maximum  value   Rm   when  r  =  a,  and  thus  we 
have 
m      a(b  —  a) 
If  we  suppose  that  b  and  V  are  fixed  and  a  is  a  variable, 
we  see  that  Rm  increases  from  a  =  0  to  a  =  b/2  and  diminishes 
for  greater  values  of  a.  Hence,  if  a  be  greater  than  b/2  and 
wre  gradually  increase  V,  the  moment  the  electric  intensity 
attains  a  certain  value  the  air  immediately  in  contact  with 
the  inner  sphere  breaks  down  and  becomes  conducting.  The 
electric  intensity  at  the  surface  of  this  stratum  of  conducting 
air  round  the  inner  sphere  will  be  greater  than  the  old 
maximum  electric  intensity,  and  hence  a  new  stratum  will 
be  broken  down.  It  is  unlikely  that  the  boundaries  of  the 
strata  successively  broken  down  will  be  exactly  spherical, 
but  any  lack  of  symmetry  will  accelerate  the  discharge  and 
an  arc  between  the  two  spheres  will  certainly  be  established, 
Thus  when  a  is  greater  than  b/2  the  sparking  voltages 
between  the  two  spheres  may  be  used  to  calculate  Rmax.  the 
dielectric  strength  of  air. 
On  the  other  hand,  if  a  be  less  than  5/2,  an  increase  in  its 
value  will  diminish  ROT,  and.  thus  equilibrium  is  possible  with 
a  conducting  stratum  of  air  round  the  inner  sphere.  The 
outside  of  this  stratum  is  what  we  call  the  corona.  As  the 
voltage  Y  is  increased  the  corona  grows  until  its  radius  is 
nearly  equal  to  b/2,  when  a  disruptive  discharge  will  ensue. 
We  see  therefore  that  the  size  of  the  inner  sphere  has  no 
practical  effect  on  the  disruptive  voltage  provided  that  its 
radius  be  less  than  b/2. 
When  the  radius  is  greater  than  b/2  we  should  expert  no 
luminous  effects  until  the  final  discharge  took  place.  This 
would  occur  at  the  instant  when 
b  —  a     a 
where  e  represents  the  lost  volts. 
4.   The  Electric  Intensity  inside  a  Concentric  Main. 
Let  us  now  consider  the  important  practical  case  of  a  con- 
centric main.  A  hollow  conducting  cylinder  of  inner  radius 
b  contains  a  coaxial  conducting  cylinder  of  radius  a.     If  the 
R  2 
