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Magnetism

Question

A magnetized needle of magnetic moments $4.8 \times 10^{- 2} A m^{2}$ is placed at $30^{0}$ with the direction of a uniform magnetic field of $3 \times 10^{- 2} T$ . If the needle is pivoted through its center of mass and is free to rotate in the plane of the magnetic field. Find the angular frequency (in rad/s) for small oscillations. The moment of inertia of the needle about the axis of rotation is $2.25 \times 10^{- 5} k {gm}^{2}$

Moderate

Solution

Time Period for small oscillations is ,
$T = 2 π \sqrt{\frac{I}{M B_{H}}}$
$\Rightarrow \frac{2 π}{T} = \sqrt{\frac{M B_{H}}{I}} \Rightarrow ω = \sqrt{\frac{M B_{H}}{I}}$
Given
$\begin{matrix} M = 4.8 X 10^{- 2} {Am}^{2} \\ B_{H} = 3 \times 10^{- 2} T \\ I = 2.25 \times 10^{- 5} {Kgm}^{2} \end{matrix}$
$\begin{array}{l} ∴ ω = \sqrt{\frac{4.8 \times 10^{- 2} \times 3 \times 10^{- 2}}{2.25 \times 10^{- 5}}} \\ ω = 8 rad / \sec \end{array}$

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