Question

A short bar magnet in a vibrating magnetometer makes $16$ oscillations in $4$ seconds. Another short magnet with same length and width having moment of inertia $1.5$ times the first one is placed over the first magnet and oscillated. Neglecting the induced magnetization, the time period of the combination is:
(A) $2\sqrt {10} \,{\text{s}}$
(B) $20\sqrt {10} \,{\text{s}}$
(C) $\dfrac{2}{{\sqrt {10} }}\,{\text{s}}$
(D) $\dfrac{{2.5}}{{\sqrt {10} }}\,{\text{s}}$

Answer 1

First of all, we will make use of the magnetic torque formula. After that we will find the moment of inertia of the second one using the first one. Then we will find the ratio of the time periods. We will manipulate accordingly and obtain the result.

Complete step by step answer:
The magnetic torque is proportional to the magnet moment and the magnetic field angle.
So,
$\tau = {\rm I}{\omega ^2}\theta = {\rm B}{\rm M}\theta$
Here,
$\omega \propto \sqrt {\dfrac{{\text{M}}}{{\text{I}}}}$
And
${\text{T}} \propto \sqrt {\dfrac{{\text{I}}}{{\text{M}}}}$
As the other surface is kept over with the second magnet.
So
${{\text{I}}_2} = \left( {1.5 + 1} \right){{\text{I}}_1} \\\ \Rightarrow{{\text{I}}_2} = 2.5\,{\text{I}} \\\$
And
${{\text{M}}_2} = 2\,{{\text{M}}_1} \\\ \Rightarrow{{\text{M}}_2} = 2\,{\text{M}} \\\$
Here,
$\dfrac{{\text{T}}}{{{{\text{T}}_0}}} = \dfrac{{\text{T}}}{{0.25}} \\\ \Rightarrow\dfrac{{\text{T}}}{{{{\text{T}}_0}}} = \sqrt {\dfrac{{2.5\,{\text{I}}}}{{2{\text{M}}}}} \sqrt {\dfrac{{\text{M}}}{{\text{I}}}} \\\$
So,
${\text{T}} = \dfrac{1}{8}\sqrt 5 \,{\text{s}}$
Or
$\therefore{\text{T}} = \dfrac{2}{{\sqrt {10} }}\,{\text{s}}$

Hence, the required answer is $\dfrac{2}{{\sqrt {10} }}\,{\text{s}}$ .The correct option is C.

Additional information:
Magnetic field and: A magnetic field is a vector field that describes the magnetic effects of other moving charges and magnetic materials on electrical charges. A charge that is travelling throughout a magnetic field has a force perpendicular to the magnetic field and its own pace. Magnetic fields are typically found in permanent magnets, which attract and repel other magnets, using magnetic materials such as iron.
Magnetic moment: The magnetic moment is the force of a magnet or another object that creates a magnetic field and its magnet orientation. Examples are: electric currents, permanent magnets, moving elementary particles (such as electrons), some molecules, and a large number of astronomical objects (e.g. several planets, several asteroids, stars, etc.) which have magnet moments.

Note: A magnet's magnetic moment is a quantity that defines the force of electric currents the magnet may apply and the torque of the magnetic field. Magnetic field lines are lines in a magnetic field whose tangent would show the direction of the field at every point and the magnitude of the field is determined by their density.