Question

Two magnets are held together in a variation magnetometer and are allowed to oscillate in the earth's magnetic field with like poles together. 12 oscillations per minute are made but unlike poles together only 4 oscillations per minute are executed. The ratio of their magnetic moments is
A. $1:3$
B. $3:1$
C. $2:3$
D. $5:4$

Answer 1

Hint: For both similar and unlike pole arrangements, the oscillation time period is determined. The oscillations of time are numerous. A magnetic object's magnetic strength and orientation are indicated by a quantity known as the magnetic moment.

Formula used:
The following formula is used to compute ratio:
$\dfrac{{{M_1}}}{{{M_2}}} = \dfrac{{{T_1}^2 + {T_2}^2}}{{{T_1}^2 - {T_2}^2}}$
Where, ${T_1}$ represents the time period when the magnets are in pole arrangements.
${T_2}$ represents the time period when the magnets are in unlike pole arrangements.

Complete step by step solution:
An instrument called a "Vibration Magnetometer" is used for the measurements of magnetic fields of 1-30,000 gauss. The electromotive force created in a tiny search coil, which vibrates with known amplitude and frequency in the magnetic field to be measured, is what drives the magnetometer action. After being magnified by a valve amplifier, the electromotive force is measured using a rectifier-type voltmeter that is directly calibrated in gauss.

A vector quantity called an object's magnetic moment explains the magnetic field the thing produces. The magnetic moment of a magnet or other item that generates a magnetic field is its magnetic pull and orientation. The passage of electricity or the spin angular momentum can both generate magnetic moments..

The information provided to us is as follows:
When the opposite poles are connected, the period of one oscillation equals
${T_1} = \dfrac{1}{{{n_1}}}$
$ \Rightarrow {T_1} = \dfrac{{60}}{{12}} = 5\sec $
Similarly
${T_2} = \dfrac{{60}}{4} = 15\sec $
To find the ratio of magnetic moment:
$\dfrac{{{M_1}}}{{{M_2}}} = \dfrac{{{{15}^2} + {4^2}}}{{{{15}^2} - {4^2}}}$
$ \Rightarrow \dfrac{{{M_1}}}{{{M_2}}} = \dfrac{5}{4}$

Hence option D is correct.

Note: If the +ve and -ve signs in the numerator and denominator are switched, you might opt for option (d). Also keep in mind ${T_1}$ represents the time period when the magnets are in like pole arrangements and ${T_2}$ represents the time period when the magnets are in unlike pole arrangements. If they are interchanged you can end up with the wrong solution.