Current Electricity Question 11
Question 11 - 30 January - Shift 1
The charge flowing in a conductor changes with time as $Q(t)=\alpha t-\beta t^{2}+\gamma t^{3}$. Where $\alpha, \beta$ and $\gamma$ are constants. Minimum value of current is :
(1) $\alpha-\frac{3 \beta^{2}}{\gamma}$
(2) $\alpha-\frac{\gamma^{2}}{3 \beta}$
(3) $\beta-\frac{\alpha^{2}}{3 \gamma}$
(4) $\alpha-\frac{\beta^{2}}{3 \gamma}$
Show Answer
Answer: (4)
Solution:
Formula: Instantaneous current
$ \begin{aligned} & Q=(\alpha t-\beta t^{2}+\gamma t^{3}) \\ & i=\frac{dQ}{dt}=(\alpha-2 \beta t+3 \gamma t^{2}) \\ & \frac{di}{dt}=(3 \gamma t-2 \beta)=0 \\ & \Rightarrow t=\frac{\beta}{3 \gamma} \\ & i=(\alpha-2 \beta t+3 \gamma t^{2})=(\alpha-\frac{\beta^{2}}{3 \gamma}) \end{aligned} $
