Assume the magnetic field

$$\mathbf{B}(\mathbf{x})=b(\mathbf{x}-3 \hat{\mathbf{z}} \hat{\mathbf{z}} \cdot \mathbf{x}),$$

where $\hat{\mathbf{z}}$ is a unit vector in the vertical direction. Show that this satisfies the expected equations for a static magnetic field in vacuum.

A circular wire loop, of radius $a$, mass $m$ and resistance $R$, lies in a horizontal plane with its centre on the $z$-axis at a height $z$ and there is a magnetic field given by $(*)$. Calculate the magnetic flux arising from this magnetic field through the loop and also the force acting on the loop when a current $I$ is flowing around the loop in a clockwise direction about the $z$-axis.

Obtain an equation of motion for the height $z(t)$ when the wire loop is falling under gravity. Show that there is a solution in which the loop falls with constant speed $v$ which should be determined. Verify that in this situation the rate at which heat is generated by the current flowing in the loop is equal to the rate of loss of gravitational potential energy. What happens when $R \rightarrow 0$ ?