Question

The restoring couple for a magnet oscillating in the vibration magnetometer is provided by
(A) The horizontal component of the earth’s magnetic field
(B) Gravity
(C) Torsion in the suspended thread
(D) The magnetic field of the magnet

Answer 1

Hint
Using the formula for calculating the time period or the frequency of the vibration magnetometer the answer can be found out. The component that helps the magnetometer to align back, if in case gets displaced will be the source of restoring force.
Formula Used: In the solution we will be using the following formula,
$\Rightarrow T = 2\pi \sqrt {\dfrac{I}{{M{B_H}}}} $
where $T$ is the time period, $M$ is the magnetic moment, $I$ is the moment of inertia and ${B_H}$ is the horizontal component of the earth’s magnetic field.

Complete step by step answer
The components of the vibration magnetometer which are dependent and independent of the external parameters are given by the following formulas of time period and frequency
$\Rightarrow \nu = \dfrac{1}{{2\pi }}\sqrt {\dfrac{{M{B_H}}}{I}} $ and,
$\Rightarrow T = 2\pi \sqrt {\dfrac{I}{{M{B_H}}}} $
Where $\nu $ is the frequency of the vibration magnetometer.
For example, when a bar magnet of the magnetometer is cut into n equal parts, the magnetic moment becomes $\dfrac{M}{n}$, the moment of inertia becomes $\dfrac{I}{n}$. But, there will be no change in the time period of vibration. Thus, we have the expression in terms of the time period of the vibration magnetometer.
$\Rightarrow T = 2\pi \sqrt {\dfrac{I}{{M{B_H}}}} $
From here we can write,
$\Rightarrow T \propto \dfrac{1}{{{B_H}}}$
In terms of the frequency of the vibration magnetometer, the expression can be written as $\nu \propto {B_H}$. The torque acts on the vibration magnetometer when it gets displaced from the earth’s magnetic field, to align it back with the earth’s magnetic field.
$\therefore $ The restoring couple for a magnet oscillating in the vibration magnetometer is provided by the horizontal component of the earth’s magnetic field.
Thus, option (B) is correct.

Note
The earth’s magnetic field is also called the geomagnetic field and it extends several kilometers in space. The north and south magnetic poles of the earth are located at roughly the geographic north and south poles. This magnetic field of the earth interacts with the solar wind and other charged particles which are coming from the sun.