Electromagnetism | Part IB, 2004

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

MX¨=BR(X+2)ddt(XB)M \ddot{X}=-\frac{B}{R(X+2 \ell)} \frac{d}{d t}(X \ell B)

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