Download Oscillations of the bar magnet

Small oscillations of the bar magnet
A small bar magnet suspended in a magnetic field will align itself with that field. A
small displacement will set it oscillating in a simple harmonic motion about the
equilibrium position. This motion is similar to a pendulum’s oscillation (in fact, this is a
torsion pendulum) and neglecting friction we can say that the total mechanical energy is
conserved during this motion.
The initial potential energy of this pendulum is Ui   B cos max , where  is
magnetic moment of the bar magnet, B is the magnitude of the external magnetic field,
and  max is the initial displacement angle. The initial kinetic energy is zero. As the
pendulum passes through equilibrium, its potential energy has its minimum, which is
U f   B (since   0 when the magnet is aligned with the field). At the same time
 2 


 T  , where I is the moment of
2
2
kinetic energy has its maximum, EK max
2
I max


2
I
2
max
inertia of the bar magnet,  max is the maximum angular speed of the magnet, and T is the
period of oscillations. Conservation of energy relates the filed, B, to the period
2

  2 2
I max
B


 2 1  cos  max    T 
Since measuring all the quantities in the square bracket is very difficult we will consider
this square bracket to be a single, unknown quantity which does not change. If we then
increase the field by a known amount, Bc , so that the new field is B  Bc , we can measure
a new period, T . The equation now becomes
2

  2 
I max
B  Bc  


 2 1  cos  max    T 
2
If we then decrease the field by the same known amount, Bc , so that the new field is
B  Bc , we can measure a new period, T . The equation then becomes
2

  2 
I max
B  Bc  


 2 1  cos  max    T 
2
The additional magnetic field Bc can be created by a current coil and its magnitude
can be calculated from
Bc 
N 0 i
,
2R
where i is the current in the coil, 0  4 107 N A2 , N is the number of turns of the
wire in the coil, and R is the radius of the coil.
Combine the last three equations to eliminate the unknown term in the square
brackets and solve for magnetic field of the Earth B. Obtain several data points with
different values of current and number of turns in the coils.