Question #43548

A rigid circular loop of radius $R = 0.1 \text{ m}$ and mass $m = 0.5 \text{ kg}$ is connected to a central hub by massless rigid spokes. The hub is suspended by a massless vertical spring of spring constant $k = 16 \text{ N/m}$, constraining the loop to move only along the vertical $z$-axis. The system is placed in an axisymmetric magnetic field given by $\vec{B} = -\alpha z r \hat{r} + (B_0 + \alpha z^2) \hat{z}$, where $r$ is the radial distance from the $z$-axis, $\hat{r}$ and $\hat{z}$ are unit vectors in cylindrical coordinates, $B_0 = 1.0 \text{ T}$, and $\alpha = \frac{45}{\pi} \text{ T/m}^2$. The equilibrium position of the loop in the absence of any current is at $z = 0$. A steady current $I = 10 \text{ A}$ is then established in the loop, flowing in the clockwise direction when viewed from the positive $z$-axis. If the loop is given a small vertical displacement from its equilibrium position, what is the period of its simple harmonic oscillations?