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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 the conductor temperature $T$ assuming the thermal power dissipated into surrounding space to vary $q = k (T - T_{0})$ , where $k$ is a constant, $T_{0}$ is the surrounding temperature (equal to conductor temperature at the initial moment).

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$T = T_{0} + \frac{V^{2}}{k R} (1 + e^{- k t / c})$

$T = T_{0} + \frac{V^{2}}{k R} (1 - e^{- k t / c})$

$T = T_{0} - \frac{V^{2}}{2 k R} (1 - e^{- k t / c})$

$T = T_{0} - \frac{2 V^{2}}{k R} (1 - e^{k t / c})$

answer is B.

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Detailed Solution

$\frac{V^{2}}{R} = k (T - T_{0}) + C (\frac{d T}{d t})$

or $\frac{d T}{\frac{V^{2}}{R} - k (T - T_{0})} = \frac{d t}{C}$

$ω \int_{T_{0}}^{T} \frac{d T}{\frac{V^{2}}{R} - k (T - T_{0})} = \int_{0}^{t} \frac{d t}{C}$

(at $t = 0$ , temperature of conductor $T = T_{0}$ )

Solving this equation, we get

$T = T_{0} + \frac{V^{2}}{R} (1 - e^{- k t / c})$

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