Question

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

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

Answer 1

The correct option is A $\frac{a^{3}R}{6b}$

$\text{Step 1: Current as function of time}$

$i=\frac{dQ}{dt}=\frac{d}{dt}(at-b{t}^2)$

$i=a-2bt$

$\text{Step 2: Time up to which current will flow}$

Heat will be produced till the time current is non-zero.

$i=a-2bt$

$0=a-2b{t}_0\Rightarrow t_0=\frac{a}{2b}$

$\text{Step 3: Heat produced from time}$ $t=0$ $\text{to}$ $t=t_0$

$H={\int }_0^{t_0}{i}^{2}Rdt$

$={\int }_0^{t_0}(a-2bt{)}^{2}Rdt$

$=R{\int }_0^{t_0}(a^2-4abt+4b^2{t}^2)dt$

$=R{[a^{2}t-2ab{t}^2+\frac{4b^2}{3}{t}^3]}_0^{t_0=\frac{a}{2b}}$

$=R[a^2\left(\frac{a}{2b}\right)-2ab{\left(\frac{a}{2b}\right)}^2+\frac{4b^2}{3}{\left(\frac{a}{2b}\right)}^3]$

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