The charge flowing in a conductor changes with time as where , and are constants. Minimum value of current is [2023]

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Solution

Q.
The charge flowing in a conductor changes with time as $Q (t) = α t - β t^{2} + γ t^{3},$ where $α$ , $β$ and $γ$ are constants. Minimum value of current is [2023]

1 $α - \frac{3 β^{2}}{γ}$

2 $α - \frac{γ^{2}}{3 β}$

3 $β - \frac{α^{2}}{3 γ}$

4 $α - \frac{β^{2}}{3 γ}$

Ans.
(4)

$Q = (α t - β t^{2} + γ t^{3})$

$i = \frac{d Q}{d t} = (α - 2 β t + 3 γ t^{2})$

$\frac{d i}{d t} = (3 γ t - 2 β) = 0 \Rightarrow t = \frac{β}{3 γ}$

$i = (α - 2 β t + 3 γ t^{2}) = (α - \frac{β^{2}}{3 γ})$

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