E & M Qual Problems

1.) A rigid, infinitesimally thin conducting ring of radius [math]\displaystyle{ a=0.15 m }[/math] and mass [math]\displaystyle{ m=1.0kg }[/math] is initially oriented nearly vertically in a uniform vertical magnetic field [math]\displaystyle{ \vec{B} = B_0 \hat{z} }[/math] where [math]\displaystyle{ B_o = 2.5 Telsa }[/math]. The resistance of the ring is [math]\displaystyle{ R = 10^{-3} \Omega }[/math]. The ring is initially at rest at a polar angle of [math]\displaystyle{ \theta_0 = 1.0 mrad }[/math] with respect to the [math]\displaystyle{ \hat{z} }[/math]-axis. The orientation of the rin is such that the unit normal vector [math]\displaystyle{ \hat{n} }[/math] lies in the [math]\displaystyle{ \hat{y} -\hat{z} }[/math] plane, initially nearly parallel to the [math]\displaystyle{ -\hat{y} }[/math] direction.

If the ring is allowed to fall with no slipping under the influence of the earth's gravitational field, the presence of the magnetic field acts as a magnetic "brake", slowing the fall of the ring by a considerable amount, as if the ring was falling in a highly viscous fluid.

a.) Calculate the instantaneous EMP [math]\displaystyle{ \epsilon(t) }[/math] induced in the ring in terms of the polar angle [math]\displaystyle{ \theta }[/math] and the angular velocity [math]\displaystyle{ \dot{\theta} }[/math]