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:

The charge flowing in a conductor is $$Q(t) = \alpha t - \beta t^2 + \gamma t^3$$.

$$I(t) = \frac{dQ}{dt} = \alpha - 2\beta t + 3\gamma t^2$$

$$\frac{dI}{dt} = -2\beta + 6\gamma t = 0$$

$$t = \frac{\beta}{3\gamma}$$

Since $$\frac{d^2I}{dt^2} = 6\gamma > 0$$ (assuming $$\gamma > 0$$), this is indeed a minimum.

$$I_{\min} = \alpha - 2\beta \cdot \frac{\beta}{3\gamma} + 3\gamma \cdot \frac{\beta^2}{9\gamma^2}$$

$$= \alpha - \frac{2\beta^2}{3\gamma} + \frac{\beta^2}{3\gamma}$$

$$= \alpha - \frac{\beta^2}{3\gamma}$$

The minimum value of current is $$\alpha - \frac{\beta^2}{3\gamma}$$.

The correct answer is Option 4: $$\alpha - \frac{\beta^2}{3\gamma}$$.