First slide

Thermal effect of current

Question

The change following through a resistance R varies with time ‘t’ as ${Q = at - bt}^{2}$ . The total heat produced in R is

Easy

A

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

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

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

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

Solution

${Q = at - bt}^{2} ⇒i = \frac{d Q}{d t} = a - 2bt n o w, i = 0 at t = \frac{a}{2b} T o t a l h e a t p r o d u c e d i s g i v e n b y H = \int_{0}^{a / 2 b} i^{2} R d t = R \int_{0}^{a / 2 b} {(a - 2 b t)}^{2} d t = R \int_{0}^{a / 2 b} (a^{2} - 4 a b t + 4 b^{2} t^{2}) d t = R {[a^{2} t - 4 a b \frac{t^{2}}{2} + \frac{4 b^{2} t^{3}}{3}]}_{0}^{a / 2 b} = R [a^{2} . \frac{a}{2 b} - \frac{4 a b}{2} . \frac{a^{2}}{4 b^{2}} + \frac{4 b^{2}}{3} . \frac{a^{3}}{8 b^{3}}] = \frac{a^{3} R}{b} [\frac{1}{2} - \frac{1}{2} + \frac{1}{6}] = \frac{a^{3} R}{6 b}$

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