\sect{Sweep of electric field over an oscillator}

Consider a massive disc of area $A$ and mass $M$ that is attached to a spring that has an elastic constant $\alpha$. The system is immersed in gas of particles that have mass $\mass$, density $n$ per unit volume, and temperature $T$. A time dependent slowly varying electric field $\mathcal{E}(t)$ is applied. The units are chosen such that the electric charge of the disc is unity.  \\

(1) Write an expression for the position  $\braket{x}$ of the disc at a given moment when the electric field has strength $\mathcal{E}$ and rate of change $\dot{\mathcal{E}}$. \\


(2) Discuss possible refinement of your result assuming that the spring has temperature $T_A$, while the gas has temperature $T_B$. Note: there are two possible perspectives for the analysis: (a)~Treating A+B as one system. (b)~Treating A as the system, and B as the bath.  
In the latter context, explain the limitations of the Langevin formulation. In the former context note that the temperature is ill defined.     

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