Question #43393

Consider two independent physical systems in a vacuum:

System I: A thick spherical shell of inner radius $R$ and outer radius $2R$ has a non-uniform mass density $\rho(r) = \rho_0 \frac{r}{R}$, where $r$ is the radial distance from the center and $\rho_0$ is a constant in $\text{kg m}^{-3}$.

System II: A very long thick-walled cylindrical solenoid of inner radius $R$ and outer radius $2R$ carries a non-uniform azimuthal volume current density $J(r) = J_0 \frac{r}{R}$, where $r$ is the radial distance from the central axis and $J_0$ is a constant in $\text{A m}^{-2}$.

Let $E_g$ be the magnitude of the gravitational field at a distance $r = R/2$ from the center of the spherical shell. Let $B$ be the magnitude of the magnetic field at a distance $r = R/2$ from the central axis of the solenoid. Which of the following correctly gives $E_g$ and $B$?