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 35 - AC Circuits - Exercises and Problems - Page 1053: 43

Answer

${\bf 44.1}\;\rm Hz$

Work Step by Step

Recalling that the voltage across a discharging capacitor is given by $$V_C=V_0e^{-t/RC}$$ And we know that $V_C=\frac{1}{2}V_0$ at $t_1=2.5$ ms, so $$\frac{1}{2} \color{red}{\bf\not} V_0= \color{red}{\bf\not} V_0e^{-t/RC}$$ $$\ln\left[\frac{1}{2}\right]=\dfrac{-t_1}{RC}$$ Hence, $$RC=\dfrac{-t_1}{\ln\left[\frac{1}{2}\right]}\tag 1$$ The crossover frequency is given by $$f_c=\dfrac{\omega_c}{2\pi}$$ where $\omega_c=1/RC$; $$f_c=\dfrac{1}{2\pi RC}$$ Plug from (1), $$f_c=\dfrac{-\ln\left[\frac{1}{2}\right]}{2\pi t_1}$$ Plug the known; $$f_c=\dfrac{-\ln\left[\frac{1}{2}\right]}{2\pi (2.5\times 10^{-3})}$$ $$f_c=\color{red}{\bf 44.1}\;\rm Hz$$
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