Question

A bar magnet of moment of inertia \[9\times {{10}^{-3}}kg{{m}^{2}}\] placed in a vibration magnetometer and oscillating in a uniform magnetic field \[16{{\pi }^{2}}\times{{10}^{-5}}T\] makes 20 oscillations in 15 s. The magnetic moment of the bar magnet is

\[\begin{align}
  & A.300A{{m}^{2}} \\
 & B.200A{{m}^{2}} \\
 & C.500A{{m}^{2}} \\
 & D.600A{{m}^{2}} \\
 & E.400A{{m}^{2}} \\
\end{align}\]

Answer 1

Hint: If there are number of oscillations in a given time period, then we have to find time period for one oscillation and apply the formula of time period for a bar magnet vibrating in given magnetic field that is given by \[T=2\pi \sqrt{\dfrac{I}{mB}}\].

Formula Used:

\[T=2\pi \sqrt{\dfrac{I}{mB}}\]

Complete step by step answer:
In our question, we are given with:

Moment of inertia of bar magnet \[=9\times {{10}^{-3}}kg{{m}^{2}}\]
Magnetic Field \[=16{{\pi }^{2}}\times {{10}^{-5}}T\]
Time period for 20 oscillations = 15 second

And we have to find the magnetic moment of the bar magnet.

Firstly we will find the time period for one oscillation.

20 oscillations = 15 seconds
1 oscillation = \[\dfrac{15}{20}\] seconds
Time Period of oscillation due magnetic field is given by:
\[T=2\pi \sqrt{\dfrac{I}{mB}}\]

Where:

\[T=\]\[\dfrac{15}{20}\] seconds
\[I=\]moment of inertia
\[m=\]magnetic moment of bar magnet
\[B=\]given magnetic field

\[\begin{align}
  & T=2\pi \sqrt{\dfrac{9\times {{10}^{-3}}}{16{{\pi }^{2}}\times {{10}^{-5}}\times m}} \\
 & \dfrac{15}{20}=2\pi \sqrt{\dfrac{9\times {{10}^{-3}}}{16{{\pi }^{2}}\times {{10}^{-5}}\times m}} \\
 & \dfrac{3}{4}=2\pi \sqrt{\dfrac{9\times {{10}^{-3}}}{16{{\pi }^{2}}\times {{10}^{-5}}\times m}} \\
 & \dfrac{3}{4}=2\pi \times \dfrac{3\times 10}{4\pi \times \sqrt{m}} \\
 & \sqrt{m}=20 \\
 & m=400A{{m}^{2}} \\
\end{align}\]

Thus the magnetic moment associated with this bar magnet is \[=400A{{m}^{2}}\]

Additional information:
Let us know the bar magnet :
a.)Bar Magnet: A bar magnet is a rectangular piece made up of ferromagnetic material. It has the North Pole and South Pole. The North Pole is aligned to the geometric North and South Pole to geometric South.
b.)If you break the magnet into two or more parts it will again assemble the North Pole and South Pole.
c.)We experience the strongest magnetic force at poles of the magnet.
d.)When the bar magnet is suspended freely it will align itself to the north-south position and finally come to rest.
e.)If we place bar magnets near to each other, the North Pole repels the North Pole of another bar magnet and the South Pole to another South Pole. But the North Pole and South Pole of another bar magnet attract each other.
f.)We can say that like poles attract, while unlike poles repel.


Note: Students must remember that we have to first find the time period for one oscillation then we have to proceed further. If you directly put time in this equation, the answer you will get will be different.