319


Efm  x  BO  —  S-ml  cos  6.  Equating  the  two  moments
and

96ml  sin  6  =  &ml  cos  0

or  &=  96  tan  6.
The  magnetic  force  producing  a  deflection  is  equal  to
the  product  of  the  strength  of  field  and  the  tangent  of  the
angle  of  deflection.
270.  Method  of  Oscillations.  —  When  a  suspended
magnetic  needle  is  disturbed  from  its  position  of  equilib-
rium it  describes  a  series  of  oscillations  like  a  pendulum.
If  the  angular  deflection  be  small  the  vibrations  will  all
be  accomplished  in  the  same  period.  The  law  of  the
vibration  of  such  a  needle  is  the  same  as  that  of  the  pen-
dulum (I.,  71),  since  the  restoring  couple  is  proportional
to  the  sine  of  the  angle  of  deflection  6  (Fig.
144) ;  and  when  this  angle  is  small  the  mo-  Nj
tion  is  simple  harmonic.  I.
We  may  therefore  write  for  the  period  of  a
complete  vibration

where  K  is  the  moment  of  inertia  of  the  needle,
96  the  intensity  of  the  field,  and  dlb  is  the
product  ml  corresponding  to  Mh  in  the  case
of  the  pendulum.
From  this  equation


*"*\li


m96  =  —=irsKn't

mi


00


or  the  intensity  of  the  field  is  proportional  to  the  square  of
the  vibration-frequency.
The  fields  at  two  places  may  be  compared  by  observing


320  ELECTRICITY    AND    MAGNETISM.
the  number  of  vibrations  made  by  the  same  magnet  in
equal  times,  first  at  the  one  place  and  then  at  the  other.
Then

86  _  T'-_n2
-  d6'~¥~nn'

271.  Comparison  of  Pole-strengths  by  Oscillations.
