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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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