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The charge Q flowing through a resistance R varies with time t as $Q = a t - b t^{2}$ . When a and b are positive constants , then the total heat produced in R upto the instant when instantaneous current becomes zero is. .......

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

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

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

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

answer is D.

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Detailed Solution

$Q = a t - b t^{2} \Rightarrow i = \frac{d Q}{d t} = a - 2 b t$

for maximum value of t

$\frac{d Q}{d t} = 0 \Rightarrow t = \frac{a}{2 b}$

$H = \int_{0}^{t} i^{2} R . d t$

$H = \int_{0}^{t} {(a - b t)}^{2} R . d t$

$H = \frac{a^{3} R}{6 b}$

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