Answer

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**Hint:**The ratio of the magnet's moment of inertia to the product of its magnetic moment and the ambient magnetic field is the square root of the time period of the vibration magnetometer. The magnetic fields for the configurations utilising magnets with various moments will not change to determine the ratio of time periods.

**Formula used:**

The time period of vibration magnetometer is,

$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 (horizontal component)

**Complete step by step solution:**

A vibration magnetometer is a device used to measure the horizontal component of the Earth's magnetic field or to compare the magnetic moments of two magnets. When a magnet hanging in a consistent magnetic field (such the one caused by the earth's magnetic field) is moved out of its equilibrium position and starts to vibrate merely harmonically about the direction of the field.

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.

The time period of vibration magnetometer is given by,

$T = 2\pi \sqrt {\dfrac{I}{{mB}}} $

So, the time period vibration magnetometer will become infinity when the magnetic field value will become zero. The only place where the magnetic field becomes zero is at the poles. There is no horizontal geomagnetic field strength in the magnetic poles of the earth.

So, at poles $B = 0$

$\therefore T = \infty $

Hence the time period will be infinite at the poles.

**Therefore, option B is the correct answer.**

**Note:**We must remember that the quantity of magnetic field lines affects the magnetic field's intensity. The magnetic field is stronger in the poles because the lines are higher there. However, at that point, the horizontal component is zero.

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