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

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

1. A

$11{\mathrm{i}}_{0}^{2}{\mathrm{Rt}}_{0}$

2. B

$13{\mathrm{i}}_{0}^{2}{\mathrm{Rt}}_{0}$

3. C

$17{\mathrm{i}}_{0}^{2}{\mathrm{Rt}}_{0}$

4. D

$15{\mathrm{i}}_{0}^{2}{\mathrm{Rt}}_{0}$

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Solution:

Total heat produced =

$\begin{array}{l}\underset{0}{\overset{{\mathrm{t}}_{0}}{\int }}{\left(\frac{3{\mathrm{i}}_{0}}{{\mathrm{t}}_{0}}\mathrm{t}\right)}^{2}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{Rdt}+{\left(3{\mathrm{i}}_{0}\right)}^{2}\mathrm{R}\text{\hspace{0.17em}}\left(2{\mathrm{t}}_{0}-{\mathrm{t}}_{0}\right)+{\mathrm{i}}_{0}^{2}\mathrm{R}\text{\hspace{0.17em}}\left(3{\mathrm{t}}_{0}-2{\mathrm{t}}_{0}\right)\\ =3{\mathrm{i}}_{0}^{2}{\mathrm{Rt}}_{0}\text{\hspace{0.17em}\hspace{0.17em}}+\text{\hspace{0.17em}}9{\mathrm{i}}_{0}^{2}{\mathrm{Rt}}_{0}\text{\hspace{0.17em}\hspace{0.17em}}+\text{\hspace{0.17em}}{\mathrm{i}}_{0}^{2}{\mathrm{Rt}}_{0}\text{\hspace{0.17em}\hspace{0.17em}}=\text{\hspace{0.17em}}13{\mathrm{i}}_{0}^{2}{\mathrm{Rt}}_{0}\end{array}$

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