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Question: A magnetic needle has magnetic moment of \(6.67\times {{10}^{-2}}A{{m}^{2}}\)and moment of inertia \...

A magnetic needle has magnetic moment of 6.67×10−2Am26.67\times {{10}^{-2}}A{{m}^{2}}and moment of inertia 7.5×10−6kgm27.5\times {{10}^{-6}}kg{{m}^{2}}. It performs 10 complete oscillations in 6.70s. What is the magnitude of the magnetic field?
1)0.2T b)0.05T c)0.08T d)0.01T \begin{aligned} & 1)0.2T \\\ & b)0.05T \\\ & c)0.08T \\\ & d)0.01T \\\ \end{aligned}

Explanation

Solution

Let us first find out the time period of the magnetic needle from given information. Next, using the relation between time period, magnetic moment, moment of inertia and magnetic field, we know all the terms except the value of magnetic field. Using one formula, we can find the magnetic field easily.

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

Complete step by step answer:
Let’s us first write down the given quantities,
I=7.5×10−6kgm2 M=6.67×10−2Am2 \begin{aligned} & I=7.5\times {{10}^{-6}}kg{{m}^{2}} \\\ & M=6.67\times {{10}^{-2}}A{{m}^{2}} \\\ \end{aligned}
Now, let’s calculate the time period,
For 10 oscillations, time taken is 6.70s
For 1 oscillation, time taken is
⇒t=6.7010=0.67s\Rightarrow t=\dfrac{6.70}{10}=0.67s
As we now know all the values, we can substitute them in the formula of time period,
T=2πIMB ⇒0.67=2π7.5×10−66.67×10−2×B ⇒B=4π2×7.5×10−6(0.67)2×6.67×10−2 ⇒B=0.01T \begin{aligned} & T=2\pi \sqrt{\dfrac{I}{MB}} \\\ & \Rightarrow 0.67=2\pi \sqrt{\dfrac{7.5\times {{10}^{-6}}}{6.67\times {{10}^{-2}}\times B}} \\\ & \Rightarrow B=\dfrac{4{{\pi }^{2}}\times 7.5\times {{10}^{-6}}}{{{(0.67)}^{2}}\times 6.67\times {{10}^{-2}}} \\\ & \Rightarrow B=0.01T \\\ \end{aligned}

So, the correct answer is “Option D”.

Additional Information: The magnetic moment of a body in any given magnetic field is the determination of its tendency to get arranged through a magnetic field. A magnet has two poles, north pole and south pole. Magnetic moment then can be defined as the magnetic strength and orientation of a magnet or other object that produces a magnetic field. In an electron, the magnetic moment of an electron is determined by its intrinsic properties of spin and electric charge. Moment of inertia is a quantity that is used to express the body’s tendency to resist angular acceleration, which is the sum of products of mass of each particle in the body with the square of its distance from the axis of rotation. Inertia is caused when some local field is present around the object.

Note: In the above question, we were given the time taken by the body for ten oscillations. We usually take this as the time period of the magnetic needle. But, the time period of the needle is the number of rotations per unit second. Therefore, we divided the given time by ten seconds to get the time taken for one rotation.