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

(D) $\alpha-\frac{\beta^{2}}{3 \gamma}$

(A) $\alpha-\frac{3 \beta^{2}}{\gamma}$

(B) $\alpha-\frac{\gamma^{2}}{3 \beta}$

(C) $\beta-\frac{\alpha^{2}}{3 \gamma}$

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Current Electricity

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 :

Answer: (D)

The current is

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

. Setting

\frac{d i}{d t} = - 2 β + 6 γ t = 0

locates the turning point at

t = \frac{β}{3 γ}

, and since the coefficient of

t^{2}

is positive this is a minimum. Substituting back,

i_{m i n} = α - \frac{2 β ^{2}}{3 γ} + \frac{β ^{2}}{3 γ} = α - \frac{β ^{2}}{3 γ}

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Single correct option