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Electric Power

A conductor h...

Question

A conductor has a temperature-independent resistance $R$ and a total heat capacity $C$ . At the moment $t = 0$ it is connected to a dc voltage $V$ . Find the time dependence of a conductor's temperature $R$ assuming the thermal power dissipated into surrounding space to vary as $q = k (T - T_{0})$ , where $k$ is a constant, $T_{0}$ is the environmental temperature (equal to the conductor's temperature at the initial moment).

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Solution

The equation of heat balance is
$\frac{V^{2}}{R} - k (T - T_{0}) = C \frac{d T}{d t}$
Put $T - T_{0} = ξ$
So, $C ξ + k ξ = \frac{V^{2}}{R}$ or, $ξ + \frac{k}{C} ξ = \frac{V^{2}}{C R}$
or, $\frac{d}{d t} (ξ e^{k t / c}) = \frac{V^{2}}{C R} e^{k t / c}$
or, $ξ e^{k t / c} = \frac{V^{2}}{k R} e^{k t / c} + A$
where $A$ is a constant. Clearly
$ξ = 0$ at $t = 0$ , so $A = - \frac{V^{2}}{k R}$ and hence,
$T = T_{0} + \frac{V^{2}}{k R} (1 - e^{- k t / C})$ .

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