A power transistor mounted on a finned heat sink can be modeled as a spatially isothermal object with internal heat generation and an external convection resistance.

(a) Consider such a system of mass \(m\), specific heat \(c\), and surface area \(A_{s}\), which is initially in equilibrium with the environment at \(T_{\infty}\). Suddenly, the device is energized such that a constant heat generation \(\dot{E}_{g}(\mathrm{~W})\) occurs. Show that the temperature response of the device is

\[\frac{\theta}{\theta_{i}}=\exp \left(-\frac{t}{R C}\right)\]

where \(\theta \equiv T-T(\infty)\) and \(T(\infty)\) is the steady-state temperature corresponding to \(t \rightarrow \infty ; \theta_{i}=T_{i}-T(\infty)\); \(T_{i}=\) initial temperature of device; \(R=\) thermal resistance \(1 / \bar{h} A_{s}\); and \(C=\) thermal capacitance \(m c\).

(b) A device which generates \(100 \mathrm{~W}\) of heat is mounted on an aluminum heat sink weighing \(0.35 \mathrm{~kg}\) and reaches a temperature of \(100^{\circ} \mathrm{C}\) in ambient air at \(20^{\circ} \mathrm{C}\) under steady-state conditions. If the device is initially at \(20^{\circ} \mathrm{C}\), what temperature will it reach \(5 \mathrm{~min}\) after the power is switched on?