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  • The charge flowing in a conductor changes with time as <img alt="Q(t)=\alpha t-\beta t^2+\gamma t^3" src="https://entrancecorner.oncodecogs.com/gif.latex?Q%28t%29%3D%5Calpha%20t-%5Cbeta%20t%5E

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 : 

Option: 1

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


Option: 2

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


Option: 3

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


Option: 4

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


Answers (1)

best_answer

\begin{aligned} & \mathrm{Q}=\alpha \mathrm{t}-\beta \mathrm{t}^2+\gamma \mathrm{t}^3 \\ & \mathrm{I}=\frac{\mathrm{dQ}}{\mathrm{dt}}=\alpha-2 \beta \mathrm{t}+3 \gamma \mathrm{t}^2 \\ & \frac{\mathrm{dI}}{\mathrm{dt}}=0=0-2 \beta+6 \gamma \mathrm{t} \Rightarrow \mathrm{t}=\frac{2 \beta}{6 \gamma}=\frac{\beta}{2 \gamma} \\ & \mathrm{I}_{\min }=\alpha-2 \beta\left(\frac{\beta}{3 \gamma}\right)+3 \gamma\left(\frac{\beta}{3 \gamma}\right)^2 \\ & \quad=\alpha-\frac{2 \beta^2}{3 \gamma}+\frac{\beta^2}{3 \gamma} \\ & \mathrm{I}_{\min }=\alpha-\frac{\beta^2}{3 \gamma} \end{aligned}

 

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