Question #43032

A galvanometer coil is modeled as a physical pendulum. It has a moment of inertia $I = 2 \times 10^{-4} \text{ kg m}^2$, a torsional spring constant $k = 10^{-2} \text{ N m/rad}$, $N = 100$ turns, cross-sectional area $A = 10^{-3} \text{ m}^2$, and total electrical resistance $R = 12 \text{ }\Omega$. The coil is placed in a uniform radial magnetic field of unknown magnitude $B$.

At $t=0$, the coil is at its equilibrium position ($\theta = 0$) and is given an initial angular velocity $\omega_0 = 10 \text{ rad/s}$. It swings and comes to instantaneous rest for the first time at an angle $\theta_f = 1 \text{ rad}$. The phase space trajectory of the coil is recorded, and the area under the angular velocity versus angle curve ($\omega$ vs $\theta$) from $\theta = 0$ to $\theta = 1 \text{ rad}$ is measured to be $S = 6 \text{ rad}^2/\text{s}$.

Assuming the only non-conservative force acting on the system is the magnetic damping due to the induced current in the closed coil circuit, what is the magnitude of the magnetic field $B$?