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A proton (charge $+e$) with mass $m_p$ has an intrisic angular momentum given by $\vec{L}$ and a magnetic moment given by $\vec{m} = \mu_B \vec{L}$. When the proton is placed in a uniform magnetic field of strength $B$ so that $\vec{L}$ makes an angle of $\theta$ with $\vec{B}$, the angular momentum precesses around $\vec{B}$.

a) Find the angular frequency $\omega_p$ with which the angular momentum precesses. Indicate the direction of precession on the figure above (into or out of page, as drawn).

b) For extra credit, derive $\mu_B$. One way you might proceed is to simply derive $\vec{m}$ and $\vec{L}$ separately for the proton, assuming uniform mass and charge distribution and a common angular velocity $\vec{\omega}$. A better way to proceed might be to relate $d m_z$ (along the axis of rotation) to $d L_z$ (ditto) assuming axial symmetry so that $\vec{L}$ is parallel to $\vec{\omega}$.