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

A
$11 i_{0}^{2} {Rt}_{0}$
B
$13 i_{0}^{2} {Rt}_{0}$
C
$17 i_{0}^{2} {Rt}_{0}$
D
$15 i_{0}^{2} {Rt}_{0}$

Solution:

Total heat produced =

$\begin{array}{l} \int_{0}^{t_{0}} {(\frac{3 i_{0}}{t_{0}} t)}^{2} Rdt + {(3 i_{0})}^{2} R (2 t_{0} - t_{0}) + i_{0}^{2} R (3 t_{0} - 2 t_{0}) \\ = 3 i_{0}^{2} {Rt}_{0} + 9 i_{0}^{2} {Rt}_{0} + i_{0}^{2} {Rt}_{0} = 13 i_{0}^{2} {Rt}_{0} \end{array}$

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