A block of metal is heated directly by dissipating power in the internal resistance of block. Because of temperature rise, the resistance increases exponentially with time and is given by R(t)=0.5e^2t, where t is in seconds. The block is connected across a 110 V source and dissipates 7644 J heat over a certain period of time. Calculate this period of time.

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Detailed Solution

$P = \frac{V^{2}}{R}$ . Let the required total time = T
Total heat $H = \int Pdt = \int_{0}^{T} \frac{V^{2}}{R} dt$
$\begin{array}{l} = V^{2} \int_{0}^{T} \frac{dt}{0 .5 e^{2 t}} = V^{2} \int_{0}^{T} 2 e^{- 2 t} dt \\ \Rightarrow H = - V^{2} {[e^{- 2 t}]}_{0}^{T} = V^{2} [1 - e^{- 2 T}] \\ 7644 = (110)^{2} [1 - e^{- 2 T}] \Rightarrow 0 .632 = 1 - e^{- 2 T} \\ \Rightarrow e^{- 2 T} = 0 .368 \Rightarrow e^{2 T} = 2 .715 \\ \Rightarrow 2 T = \ln (2.715) \Rightarrow T = 0 .5 s \end{array}$