Question

The charge flowing in a conductor changes with time as $Q(t)=\alpha t-\beta t ^2+\gamma^3$. Where $\alpha, \beta$ and $\gamma$ are constants. Minimum value of current is :

• A. $\alpha-\frac{\gamma^2}{3 \beta}$
• B. $\beta-\frac{\alpha^2}{3 \gamma}$
• C. $\alpha-\frac{\beta^2}{3 \gamma}$
• D. $\alpha-\frac{3 \beta^2}{\gamma}$

Answer 1

Answer: (C)

$ Q =\left(\alpha t -\beta t ^2+\gamma t ^3\right) $
$ i =\frac{ dQ }{ dt }=\left(\alpha-2 \beta t +3 \gamma t ^2\right)$
$ \frac{ di }{ dt }=(3 \gamma t -2 \beta)=0$
$\Rightarrow t =\frac{\beta}{3 \gamma} $
$ i =\left(\alpha-2 \beta t +3 \gamma t ^2\right)=\left(\alpha-\frac{\beta^2}{3 \gamma}\right)$