Question

In sum and difference method in vibration magnetometer, the time period is more if
A. Similar poles of both magnets are on same sides
B. Opposite poles of both magnets are on same sides
C. Both magnets are perpendicular to each other
D. Nothing can be said

Answer 1

Hint: In order to solve this problem, we are going to use the expression of the time period of a vibration magnetometer and then apply the sum and difference method. Upon comparing the both time periods we can arrive at the correct answer.

Formula used:
The expression of time period of vibration magnetometer is written as,
$T = 2\pi \sqrt {\dfrac{I}{{mB}}} $
Where $I = $moment of inertia about the axis of rotation
$m = $Magnetic moment of magnet
$B = $Earth's magnetic field

Complete step by step solution:
A bar magnet will rotate and attempt to align itself with the magnetic field when it is placed in a uniform magnetic field because of the force acting in the field's direction. Due to the rotational inertia caused by the torque, the magnet will rotate and attempt to align with the field, causing the bar magnet to move in a straightforward harmonic manner.

The torque acting on a bar magnet and rotational inertia are the basic operating principles of a vibration magnetometer. The time period of vibration magnetometer is given by
$T = 2\pi \sqrt {\dfrac{I}{{mB}}} $
In the sum position we can say that
${T_1} = 2\pi \sqrt {\dfrac{{{I_1}}}{{({m_1} + {m_2})B}}} $
In the difference position we can say that
${T_2} = 2\pi \sqrt {\dfrac{{{I_1}}}{{({m_1} - {m_2})B}}} $
We can clearly see that ${T_2} > {T_1}$
Hence, the time period is more if opposite poles of both magnets are on the same sides.

Therefore option B is the correct answer.

Additional information: A vibration magnetometer is a device that compares the magnetic moments of two magnets or measures the horizontal component of the Earth's magnetic field. When a magnet suspended in a constant magnetic field (such as the one created by the earth's magnetic field) is shifted out of its equilibrium position and begins to vibrate only harmonically about the field's direction.

Note: The torque and rotation in a uniform magnetic field is the basis for the operation of vibration magnetometers. The magnet will not oscillate and will stop when aligned with the field if the field is weak or not uniform. Therefore, a homogeneous, weak magnetic field is required for the vibration magnetometer to function.