Question

The value of the Earth’s magnetic field ${{B}_{H}}$=0.3 gauss. In this magnetic field a magnet is oscillating with 5oscillations/min. To increase the oscillations of the magnet up to 10oscillations/min. The value of the Earth’s magnetic field is decreased by:
a)0.3 gauss
b)0.6 gauss
c)0.9 gauss
d)0.075 gauss

Answer 1

In the above question it is asked to determine the increase in the value of the Earth’s magnetic field for the particular magnet to have an increase of 5 oscillations per minute. The value of the Earth’s magnetic field is already given to us. First we will use the expression for the time period of oscillation of a magnet in an external magnetic field in order to determine the external required magnetic field in both the cases. Further we will take the ratio of both the magnetic fields and accordingly determine the degree to which the magnetic field of the Earth needed to be increased.

Formula used: $T=2\pi \sqrt{\dfrac{I}{MB}}$

Complete step by step answer:
Let us say we have a magnet whose magnetic moment is given by ‘M’ and the moment of inertia of the magnet is given by ‘I’. If we place this magnet in an external magnetic field ‘B’ the magnet will start oscillating. In such case the time period of oscillation of the magnet is given by,
$T=2\pi \sqrt{\dfrac{I}{MB}}$
It is given that the value of the Earth’s magnetic field ${{B}_{H}}$=0.3 gauss where the magnet is oscillating with 5 oscillations/min. Using the above expression we can write,
$\begin{aligned} & T=2\pi \sqrt{\dfrac{I}{MB}} \\\ & \Rightarrow 5=2\pi \sqrt{\dfrac{I}{M(0.3)}}...(1) \\\ \end{aligned}$

Let us say the increase in magnetic field for 10 oscillations/min be B. Hence using the above equation we get,
$\begin{aligned} & T=2\pi \sqrt{\dfrac{I}{MB}} \\\ & \Rightarrow 10=2\pi \sqrt{\dfrac{I}{MB}}...(2) \\\ \end{aligned}$
Taking the ratio of equation 1 and 2 we get,
$\begin{aligned} & \dfrac{5}{10}=\dfrac{2\pi \sqrt{\dfrac{I}{M(0.3)}}}{2\pi \sqrt{\dfrac{I}{MB}}} \\\ & {{\left( \dfrac{1}{2} \right)}^{2}}=\dfrac{B}{0.3} \\\ & \Rightarrow \dfrac{1}{4}=\dfrac{B}{0.3} \\\ & \Rightarrow B=0.075gauss \\\ \end{aligned}$

So, the correct answer is “Option d”.

Note: The oscillations of a magnet in an external magnetic field depends on magnetic dipole moment, moment of inertia of the magnet and the external magnetic field. In order to increase the oscillations of the magnet we can just decrease the magnitude of the external magnetic field. It is to be noted that the oscillations of the magnet depends on the location at different places on the Earth.