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\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 
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\[\ddot{x}+\omega_0^2x = -\gamma \dot{x} + A(t) + \alpha I\]
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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, what is the interaction term $\mathcal{H}_{int}$ in the Hamiltonian, 
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.   

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