Consider a conductor in the shape of a closed curve $C$ moving in the presence of a magnetic field B. State Faraday's Law of Induction, defining any quantities that you introduce.

Suppose $C$ is a square horizontal loop that is allowed to move only vertically. The location of the loop is specified by a coordinate $z$, measured vertically upwards, and the edges of the loop are defined by $x=\pm a,-a \leqslant y \leqslant a$ and $y=\pm a,-a \leqslant x \leqslant a$. If the magnetic field is

$$\mathbf{B}=b(x, y,-2 z),$$

where $b$ is a constant, find the induced current $I$, given that the total resistance of the loop is $R$.

Calculate the resulting electromagnetic force on the edge of the loop $x=a$, and show that this force acts at an angle $\tan ^{-1}(2 z / a)$ to the vertical. Find the total electromagnetic force on the loop and comment on its direction.

Now suppose that the loop has mass $m$ and that gravity is the only other force acting on it. Show that it is possible for the loop to fall with a constant downward velocity $R m g /\left(8 b a^{2}\right)^{2}$.