91Ó°ÊÓ

$\mathrm{Ó\neg }=45\mathrm{rad}\backslash \sec$

${\mathrm{B}}_{\mathrm{h}}=16\mathrm{μ°Õ}$

$\mathrm{L}=0.04\mathrm{m}$

$\mathrm{m}=0.050\mathrm{kg}$

Here we have to use the formula for torque. Then using equations of moment of inertia and angular acceleration, we can simplify the equation of torque to equation of oscillation. Then comparing it with the general equation of torque, we get the equation of angular velocity. We can rearrange that equation for magnetic dipole moment, and using the given values, we can solve it.

Formula:

$\mathrm{τ}=\mathrm{μ}×{\mathrm{B}}_{\mathrm{h}}$

$\mathrm{Ï„}=\mathrm{\pm õÎ\pm }$

Torque due to magnetic field is given by following formula:

$\begin{array}{c}\mathrm{Ï„}=−\mathrm{μ}×{\mathrm{B}}_{\mathrm{h}}\\ =−{\mathrm{μ\mu þ}}_{\mathrm{h}}\mathrm{²õ¾\pm ²Ôθ}\end{array}$

And we know that

$\mathrm{Ï„}=\mathrm{\pm õÎ\pm }$

$−{\mathrm{μ\mu þ}}_{\mathrm{h}}\mathrm{²õ¾\pm ²Ôθ}=\mathrm{\pm õÎ\pm }$

We know that

$\mathrm{Î\pm }=\frac{{\mathrm{d}}^2\mathrm{θ}}{{\mathrm{dt}}^2}$

$\mathrm{I}=\frac{{\mathrm{mL}}^2}{12}$

$−{\mathrm{μ\mu þ}}_{\mathrm{h}}\mathrm{²õ¾\pm ²Ôθ}=\frac{{\mathrm{mL}}^2}{12}×\frac{{\mathrm{d}}^2\mathrm{θ}}{{\mathrm{dt}}^2}$

$\frac{{\mathrm{mL}}^2}{12}×\frac{{\mathrm{d}}^2\mathrm{θ}}{{\mathrm{dt}}^2}+{\mathrm{μ\mu þ}}_{\mathrm{h}}\mathrm{²õ¾\pm ²Ôθ}=0$

$\frac{{\mathrm{d}}^2\mathrm{θ}}{{\mathrm{dt}}^2}+\frac{12{\mathrm{μ\mu þ}}_{\mathrm{h}}\mathrm{²õ¾\pm ²Ôθ}}{{\mathrm{mL}}^2}=0$

$\frac{{\mathrm{d}}^2\mathrm{θ}}{{\mathrm{dt}}^2}+\frac{12{\mathrm{μ\mu þ}}_{\mathrm{h}}}{{\mathrm{mL}}^2}\mathrm{θ}=0$

Comparing this equation withtheequation of oscillation,$\frac{{\mathrm{d}}^2\mathrm{θ}}{{\mathrm{dt}}^2}+{\mathrm{Ó\neg }}^2\mathrm{θ}=0$

We get

${\mathrm{Ó\neg }}^2=\frac{12{\mathrm{μ\mu þ}}_{\mathrm{h}}}{{\mathrm{mL}}^2}$

$\begin{array}{c}\mathrm{μ}=\frac{{\mathrm{mL}}^2{\mathrm{Ó\neg }}^2}{12{\mathrm{B}}_{\mathrm{h}}}\\ =\frac{0.050×{0.04}^2×{45}^2}{12×16×{10}^{−6}}\\ =8.4×{10}^2\mathrm{J}/\mathrm{T}\end{array}$

The magnitude of magnetic dipole moment is $8.4×{10}^2\mathrm{J}/\mathrm{T}$.