First slide

Electrical power

Question

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

Difficult

A

$\frac{a^{3} R}{2 b}$
B

$\frac{a^{3} R}{b}$
C

$\frac{a^{3} R}{6 b}$
D

$\frac{a^{3} R}{3 b}$

Solution

$Given, Q = a t - b t^{2} ∴ I = \frac{d Q}{d t} = a - 2 b t$
$At t = 0, Q = 0, I = a$
$Also, I = 0 at t = a / 2 b$
Total heat produced in resistance R,
$H = \int_{0}^{a / 2 b} I^{2} R d t = R \int_{0}^{a / 2 b} (a - 2 b t)^{2} d t$
$\begin{array}{l} = R \int_{0}^{a / 2 b} (a^{2} + 4 b^{2} t^{2} - 4 a b t) d t \\ = R {[a^{2} t + 4 b^{2} \frac{t^{3}}{3} - 4 a b \frac{t^{2}}{2}]}_{0}^{a / 2 b} \end{array}$
$\begin{array}{l} = R [a^{2} \times \frac{a}{2 b} + \frac{4 b^{2}}{3} \times \frac{a^{3}}{8 b^{3}} - \frac{4 a b}{2} \times \frac{a^{2}}{4 b^{2}}] \\ = \frac{a^{3} R}{b} [\frac{1}{2} + \frac{1}{6} - \frac{1}{2}] = \frac{a^{3} R}{6 b} \end{array}$

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