A resistor is made of a material which undergoes a phase transition so that its resistance takes one of the two values, $R_{1}$ if its temperature is smaller than $T_{c}$, and $R_{2}>R_{1}$ if the temperature is larger than $T_{c}$.

This resistor is connected to a voltage source through an inductor of inductance $L$. It appears that if the applied voltage $V$ is between two critical values, $V_{1}<V<V_{2}$, the temperature of the resistor starts oscillating. Assume that (i) the heat flux $P$ from the resistor to the ambient medium is given by $P=\alpha\left(T-T_{0}\right)$, where $\alpha$ is a constant, $T$ denotes the temperature of the resistor, and $T_{0}$ is the ambient temperature; (ii) the geometrical size of the resistor is so small that it will reach a thermal equilibrium much faster than the characteristic time $L / R_{2}$.

(a)  ^2.00 Express $V_{1}$ and $V_{2}$ in terms of the other parameters defined above.

(b)  ^6.00 Assuming that $V_{1}<V<V_{2}$, sketch qualitatively how the temperature of the resistor $T$ depends on time $t$, and find the ratio $\left(T_{\text {max }}-T_{0}\right) /\left(T_{\text {min }}-T_{0}\right)$, where $T_{\text {max }}$ and $T_{\text {min }}$ denote the maximal and minimal values of $T$, respectively.

(c)  ^2.00 Find the period of oscillations if $V=\sqrt{V_{1} V_{2}}$ and $R_{2}=16 R_{1}$.