Question

The charge flowing through a resistance $R$ varies with time t as $ Q = at - bt^2$ , where $a$ and $b$ are positive constants, The total heat produced in $R$ is :

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

Answer 1

Answer: (D)

$Q = at - bt^2$
$ i = a - 2bt $ { for i = 0 $\Rightarrow \, t = \frac{a}{2b} $ }
From joule's law of heating
$dH = i^2Rdt$
$H = \int\limits^{a/2b}_0 (a - 2bt)^2 Rdt$
$H = \frac{(a - 2bt)^3 R}{ - 3 \times 2b} \Bigg|^{\frac{a}{2b}}_{0} = \frac{a^3R}{6b}$