Question

Moment of inertia of a magnetic needle is $40 \mathrm{gm}-\mathrm{cm}^{2}$ has time period 3 seconds in earth's horizontal field $=3.6 \times 10^{-5}$ weber $/ \mathrm{m}$. Its magnetic moment will be
A. $0.5~\text{A}\times {{\text{m}}^{2}}$
B. $5 \mathrm{~A} \times \mathrm{m}^{2}$
C. $0.250~\text{A}\times {{\text{m}}^{2}}$
D. $5 \times 10^{2} \mathrm{~A} \times \mathrm{m}^{2}$

Answer 1

Hint: A magnetic moment, also referred to as a magnetic dipole moment, is a measurement of an object's tendency to align with a magnetic field. A vector quantity is the magnetic moment. The magnetic moment vector frequently aligns with the magnetic field lines when the objects are positioned in that way. A magnet's magnetic moment is directed from its southern to northern poles. The magnetic field that a magnet creates is inversely related to its magnetic moment.

Formula used:
The expression of time period of a magnet in a vibration magnetometer is,
$\mathrm{T}=2 \pi \sqrt{\dfrac{\mathrm{I}}{\mathrm{MB}}}$
Here, $I$ is the moment of inertia, $M$ is the magnetic moment and $B$ is the magnetic field.

Complete step by step solution:
Given that, the moment of inertia of the needle is$40 \mathrm{gm}-\mathrm{cm}^{2}$ and the time period is 3 seconds.
Equation for time period
$\mathrm{T}=2 \pi \sqrt{\dfrac{\mathrm{I}}{\mathrm{MB}}} \\ $
$\Rightarrow I=40gm-\text{c}{{\text{m}}^{2}} \\ $
Convert gram into kilogram
$I=40 g m-\mathrm{cm}^{2}=400 \times 10^{-8} \mathrm{~kg}-\mathrm{m}^{2} \\ $
$\Rightarrow 3=2 \pi \sqrt{\dfrac{400 \times 10^{-8}}{36 \times 10^{-6} \times \mathrm{M}}} \\ $
Simplify the equation to find $\text{M}$
$\Rightarrow \dfrac{1}{\text{M}}=\dfrac{9}{4{{\pi }^{2}}}\times \dfrac{36}{4} \\ $
$\Rightarrow \dfrac{1}{\mathrm{M}}=\dfrac{9}{4 \pi^{2}} \times \dfrac{36}{4} \\
\therefore \mathrm{M}=0.5 \times \mathrm{A} \times \mathrm{m}^{2}$

Hence, the correct answer is option A.

Note: The term "moment of inertia" refers to the quantity that describes how a body resists angular acceleration and is calculated by multiplying each particle's mass by its square of distance from the rotational axis. Alternatively, it can be explained in more straightforward terms as a quantity that determines the amount of torque required for a particular angular acceleration in a rotational axis. The rotational inertia or angular mass are other names for the moment of inertia. The definition of moment of inertia often refers to a certain rotational axis. It primarily depends on how mass is distributed around a rotational axis. Depending on the axis used, MOI changes.