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Transistor

A time varyin...

Question

$A$ time varying current $i$ is passed through a resistance $R$ as shown in figure. The total heat generated in the resistance is

13 (i_{0})^{2} T t_{0}

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11 (i_{0})^{2} T t_{0}

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15 (i_{0})^{2} T t_{0}

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17 (i_{0})^{2} T t_{0}

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Solution

The correct option is A $13 (i_{0})^{2} T t_{0}$
From $t = 0 \to t_{0}$
From the graph
$i = \frac{3 i_{0}}{i_{0}} t$ (from straight line equation)

We know that, $H = i^{2} R t^{2}$
Total heat produced
$= \int_{0}^{t_{0}} (\frac{3 i_{0}}{t_{0}})^{2}$ $R d t + (3 i_{0})^{2} R (2 t_{0} - t_{0}) + (i_{0})^{2} R (3 t_{0} - 2 t_{0})$
$= 3 (i_{0})^{2} R t_{0} + 9 (i_{0})^{2} R t_{0} + (i_{0})^{2}$ $R t_{0} = 13 (i_{0})^{2} R t_{0}$
Hence, option $(b)$ is correct.

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