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(B) The total heat $H$ generated in a resistance $R$ is given by $H = \int i^2 R dt$.For the interval $0$ to $t_0$,the current is $i = \frac{3i_0}{t_0

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$13i_0^2Rt_0$

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(B) The total heat $H$ generated in a resistance $R$ is given by $H = \int i^2 R dt$.For the interval $0$ to $t_0$,the current is $i = \frac{3i_0}{t_0}t$. Heat $H_1 = \int_0^{t_0} (\frac{3i_0}{t_0}t)^2 R dt = \frac{9i_0^2 R}{t_0^2} [\frac{t^3}{3}]_0^{t_0} = 3i_0^2 Rt_0$.For the interval $t_0$ to $2t_0$,the current is constant at $i = 3i_0$. Heat $H_2 = (3i_0)^2 R (2t_0 - t_0) = 9i_0^2 Rt_0$.For the interval $2t_0$ to $3t_0$,the current is constant at $i = i_0$. Heat $H_3 = (i_0)^2 R (3t_0 - 2t_0) = i_0^2 Rt_0$.Total heat $H = H_1 + H_2 + H_3 = 3i_0^2 Rt_0 + 9i_0^2 Rt_0 + i_0^2 Rt_0 = 13i_0^2 Rt_0$.

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

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