Thermistors also exhibit memristor properties – its temperature depends on the current at previous points in time:\[C\frac{\mathrm dT}{\mathrm dt}=-\kappa(T-T_0)+VI,\]where $T_0$ is the room temperature, coefficient $\kappa$ determines the rate of heat exchange with the environment, and $C$ is the heat capacity of the thermistor. The resistance of the thermistor depends on the temperature according to the formula:\[R(T)=R_0\exp\left[\frac{E_g}{k_BT}\right],\]whete $E_g$ is activation energy (depends on thermistor's semiconducting structure), $k_B$ is Boltzmann constant.

An alternating voltage $V(t)=V_0\cos\omega t$ was applied to the thermistor, and it reached a temperature of $T_1$ on average.

A1 By averaging the equation for temperature, obtain the equation that relates $V_0$, $T_1$, and the other variables of the problem.

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The table below shows the dependence of the average temperature of the thermistor on the applied voltage. The room temperature is $t_0=20\,{}^\circ\mathrm C$, and the resistance of the thermistor at this temperature is $R=8.60~\Omega$.

A2 Calculate the activation energy $E_g$. Express your answer in electronvolts ($1~eV=1.60\cdot10^{-19}~J$). Boltzmann constatn is equal to $k_B=1.38\cdot10^{-23}~J/K$.

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A3 Calculate $R_0$ and $\kappa$.

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Let's now consider the voltage dependence of the current through the thermistor, taking into account in the first approximation the time dependence of temperature (and resistance). In the first approximation we can write:\[T(t)=T_1+T_2\cos[2\omega t+\phi],\]where $T_2\ll (T_1-T_0)$.

B1 Find expressions for $T_2$ and $\phi$.

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The figure below shows several hysteresis loops of a thermistor. Input voltage frequency is equal to:\[\nu\equiv\frac{\omega}{2\pi}=0.2~Hz.\]

B2 Calculate thermistor's heat capacity $C$.

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