The magnetic moment $\vec{\mu}$ of a spinning object with charge $Q$ and mass $M$ is proportional to its angular momentum $\vec{L}$ according to $\overrightarrow{\boldsymbol{\mu}}=g(Q / 2 M) \overrightarrow{\boldsymbol{L}}$. The dimensionless coefficient $g$ is known as the $g$ -factor of the object. Consider a spherical shell with radius $R$ and uniform charge density $\sigma$ spinning with angular velocity $\omega$ around the z-axis. (a) What is the differential area $d a$ of the ring at latitude $\theta$, with width $R d \theta,$ as shown in Fig. $P 41.57 ?$ (b) The current $d I$ carried by the ring is its charge $\sigma$ da divided by the period of rotation. Determine $d I$ in terms of $R, \sigma, \omega,$ and $\theta .(\mathrm{c})$ Determine the magnetic moment $d \overrightarrow{\boldsymbol{\mu}}$ of the ring by multiplying $d I$ by the area enclosed by the ring. (d) Determine the magnetic moment of the spherical shell by integrating over the sphere. Express your result in terms of the total charge $Q=4 \pi R^{2} \sigma$.
(e) Now consider a solid sphere of radius $R$ with volume charge density $\rho=\rho(r),$ where $r$ is the distance from the center, spinning with angular velocity $\omega$ about the z-axis. Determine its magnetic moment by integrating $\overrightarrow{\boldsymbol{\mu}}=\int d \overrightarrow{\boldsymbol{\mu}},$ where $d \overrightarrow{\boldsymbol{\mu}}$ is now the magnetic moment associated with the shell at radius $r$ with differential width $d r$. Express your answer in terms of an integral $\int_{0}^{R} r^{4} \rho(r) d r .$ (f) The angular momentum of the sphere is $\vec{L}=I \vec{\omega},$ where $I=c M R^{2}$ is its moment of inertia, and $c$ is a dimensionless factor determined by the mass distribution. Determine the $g$ -factor in terms of the $\rho$ integral and the value of $c .(g)$ If the charge is uniformly distributed so that $\rho=Q /\left(\frac{4}{3} \pi R^{3}\right)$ and if the mass is uniformly distributed so that $c=\frac{2}{5},$ then what is the value of $g ?$