Question #43850

A horizontal conducting rod of mass $m = 0.1 \text{ kg}$ and length $l = 1 \text{ m}$ is suspended by a vertical ideal spring of spring constant $k = 6 \text{ N/m}$. The rod is placed in a uniform horizontal magnetic field $B = 0.8 \text{ T}$ perpendicular to its length and the direction of motion. The ends of the rod are connected to a stationary external circuit containing a resistor $R = 1 \ \Omega$ and an inductor $L = 0.1 \text{ H}$ in series. The system is set into small vertical oscillations. The vertical displacement $x(t)$ of the rod can be expressed as a sum of a purely exponentially decaying term and an exponentially decaying oscillatory term. What is the time constant $\tau$ for the decay of the amplitude of the oscillatory term? (The amplitude of the oscillatory term decays as $A_0 e^{-t/\tau}$).