%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \sect{Galvanometer} A galvanometer can be regarded as a spring-held pointer that has mass $M$, natural oscillation frequency $\omega_0$, and a damping coefficient ${\gamma}$. The position $x$ of the spring indicates the current $I$. It obeys the equation % $\ddot{x}+\omega_0^2x = -\gamma \dot{x} + A(t) + \alpha I$ % where $A(t)$ represents an environmentally induced white noise that has a spectral intensity $\nu$, and $\alpha$ is a coupling constant. \begin {itemize} \item[(1)] On the basis of the above Langevin equation write a $d\omega$ integral for the variance $\langle x^2 \rangle$ in the absence of current. \item[(2)] Based on canonical FDT considerations deduce what is the result of the integral that you wrote in the previous item. \item[(3)] For a constant $I$, what is the average position $\langle x \rangle$ of the pointer? \item[(4)] Regarding $I$ as a driving source, write what is the conjugate variable, and what is the associate susceptibility $\chi(\omega)$. \item[(5)] Write an expression for the average rate of energy absorption $\dot{W}$, given that the current source has a frequency $\omega$ and RMS amplitude $I_{0}$. \item[(6)] The expression for $\dot{W}$ is formally the same as for a current source that is connected to a parallel RLC circuit. Write expressions for the effective values of $R$ and $L$ and $C$. \end {itemize} {\bf Tip:} The equation of a parallel RLC circuit can be written as ${G(\omega)V_{\omega}=I_{\omega}}$ where ${G(\omega)}$ is a sum of three terms. Capacitors and inductors are described by $I=C\dot{V}$ and by $V=L\dot{I}$ respectively. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%