Question

The charge flowing through a resistance R varies with time as $Q=2t-8t^2$. The total heat produced in the resistance is
$(\text{for}0\le t\le \frac{1}{8})$

• A. $\frac{R}{6}$
• B. $\frac{R}{3}$
• C. $\frac{R}{2}$
• D. $R$

Answer 1

The correct option is A $\frac{R}{6}$

We know that,
$i=\frac{dQ}{dt}=2-16t$
Heat $=R{\int }_0^{1/8}{\left(i\right)}^2.dt$
Heat $=R{\int }_0^{1/8}{(2-16t)}^2.dt$
Heat $=R{\int }_0^{1/8}(4+256t^2-64t).dt$
Heat $=R{[4t+\frac{256t^3}{3}-32t^2]}_0^{1/8}$
Heat $=R\left[4\right(\frac{1}{8})+\frac{256(\frac{1}{8}{)}^3}{3}-32(\frac{1}{8}{)}^2]$
Heat $=R\left[\right(\frac{1}{2})+\frac{1}{6}-(\frac{1}{2}\left)\right]$

Heat=$\frac{R}{6}J$