A wire is bent into the shape of three sides of a rectangle and is held fixed in the $z=0$ plane, with base $x=0$ and $-\ell<y<\ell$, and with arms $y=\pm \ell$ and $0<x<\ell$. A second wire moves smoothly along the arms: $x=X(t)$ and $-\ell<y<\ell$ with $0<X<\ell$. The two wires have resistance $R$ per unit length and mass $M$ per unit length. There is a time-varying magnetic field $B(t)$ in the $z$-direction.

Using the law of induction, find the electromotive force around the circuit made by the two wires.

Using the Lorentz force, derive the equation

$$M \ddot{X}=-\frac{B}{R(X+2 \ell)} \frac{d}{d t}(X \ell B)$$