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 :
Solution
<p>$Q(t) = \alpha t - \beta {t^2} + \gamma {t^3}$</p>
<p>$i(t) = \alpha - 2\beta t + 3\gamma {t^2}$</p>
<p>${{di} \over {dt}} = - 2\beta + 6\gamma t = 0$ (for max/min of i)</p>
<p>at $t = {\beta \over {3r}}$ (i is minimum as i is an upward parabola)</p>
<p>$$i\left( {{\beta \over {3\gamma }}} \right) = \alpha - 2\beta \left( {{\beta \over {3\gamma }}} \right) + {{3\gamma {\beta ^2}} \over {9{\gamma ^2}}}$$</p>
<p>$= \alpha {{ - {\beta ^2}} \over {3\gamma }}$</p>
About this question
Subject: Physics · Chapter: Current Electricity · Topic: Ohm's Law and Resistance
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