Aptitude Test Problems in Physics Science for Everyone by S Krotov ( PDFDrive.com )

(^248) Aptitude Test Problems in Physics
area of the hatched region is smaller than the area
of the large circle by a factor of sin (a/2). Hence
aS _^.^ a sin (a/2)
E' nR^2 sin —^
4ne 0 R 2 4neoR 2 2 4eo^
In the case of a hemisphere, a = n and
a
480
3.4. It can easily be seen from symmetry con-
siderations that the vector of the electric field
strength produced by the "lobule" with an angle a
lies in the planes of longitudinal and transverse
symmetry of the lobule. Let the magnitude of
this vector be E. Let us use the superposition prin-
ciple and complement the lobule to a hemisphere
charged with the same charge density. For this
purpose, we append to the initial lobule another
lobule with an angle n — a. Let the magnitude
of the electric field strength vector produced by
this additional lobule at the centre of the sphere
be E'. It can easily be seen that vectors E and E'
are mutually perpendicular, and their vector sum
is equal to the electric field vector of the hemi-
sphere at its centre. By hypothesis, this sum is
equal to Ea. Since the angle between vectors E
and E 0 is n/2 — a/2, we obtain
a
E= E°
sin —
2 '
3.5*. Let us consider the case when the capacitors
are oriented so that the plates with like charges

• 71
  
  
  • x+1
    72
  
  
  
  

+1 F

Fig. 208

face each other (Fig. 208). The field produced by

the first capacitor on the axis at a distance x