Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 31 - Fundamentals of Circuits - Exercises and Problems - Page 919: 71

Answer

${\bf 72.8}\;\rm \Omega$

Work Step by Step

We know that the discharge of a capacitor through a resistor is given by $$Q=Q_0e^{-t/\tau}$$ where $\tau=RC$, $$Q=Q_0e^{-t/RC}$$ Recalling that $\Delta V_C=Q/C$, so $Q_0=C\Delta V_0$, and $Q=C\Delta V$ Hence, $$\color{red}{\bf\not} C\Delta V=\color{red}{\bf\not} C\Delta V_0e^{-t/RC}$$ $$ \Delta V= Delta V_0e^{-t/RC}$$ Hence, $$\ln\left[ \dfrac{\Delta V}{\Delta V_0} \right]=\dfrac{-t}{RC}$$ Therefore, $$R =\dfrac{-t}{ C\ln\left[ \dfrac{\Delta V}{\Delta V_0} \right]}$$ Plug the given; where we can see from the given graph that at $t=0$ ms, $\Delta V_0=30$ V, and at $t=4$ ms $\Delta V=10$ V. $$R =\dfrac{-(4\times 10^{-3})}{ (50\times 10^{-6})\ln\left[ \dfrac{10}{30} \right]}$$ $$R=\color{red}{\bf 72.8}\;\rm \Omega$$
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