Question

A combination of two bar magnets, in vibration magnetometer, makes 10 oscillations per second if their like poles are tied together and 2 oscillations per second when unlike poles are tied together. If induced magnetism is neglected, then the ratio of their magnetic moments is
A. $\dfrac{3}{2}$
B.$\dfrac{13}{12}$
C.$\dfrac{8}{9}$
D.$\dfrac{12}{11}$

Answer 1

Hint: Time period of one oscillation is calculated for both like and unlike pole arrangement. Time period is one upon no. of oscillations. The magnetic moment is defined as the quantity that demonstrates the magnetic strength and orientation of any magnetic object. Ratio is calculated by using the formula:
$\dfrac{{{M}_{1}}}{{{M}_{2}}}=\dfrac{T_{1}^{2}+T_{2}^{2}}{T_{2}^{2}-T_{1}^{2}}$

Complete step by step answer:
The data given to us are as follows,
When the unlike poles are tied together, time period of one oscillation is
${{T}_{1}}=0.5s$
When the like poles are tied together, time period of one oscillation is
${{T}_{2}}=\dfrac{1}{10}s=0.1s$
To find the ratio of magnetic moment
$\dfrac{{{M}_{1}}}{{{M}_{2}}}=\dfrac{T_{1}^{2}+T_{2}^{2}}{T_{2}^{2}-T_{1}^{2}}=\dfrac{{{(0.5)}^{2}}+{{(0.1)}^{2}}}{{{(0.5)}^{2}}-{{(0.1)}^{2}}}=\dfrac{0.25+0.01}{0.25-0.01}=\dfrac{13}{12}$
The correct answer is option (b).

Additional information:
An instrument called a "Vibration Magnetometer" is used for the measurements of magnetic fields of 1-30,000 gauss. The action of the magnetometer depends on the measurement of the electromotive force induced in a small search coil which vibrates with known amplitude and frequency in the magnetic field to be measured. The electromotive force - after magnification by means of a valve amplifier - is read on a rectifier type voltmeter, calibrated directly in gauss.

Note: The possibility is that you might choose option (d) if you interchange the +ve and -ve signs in the numerator and denominator. Vibration magnetometer is used for comparison of magnetic moments and magnetic fields. This device works on the principle that whenever a freely suspended magnet in a uniform magnetic field is disturbed from its equilibrium position, it starts vibrating about the mean position.