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: 37

Answer

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Work Step by Step

$$\color{blue}{\bf [a]}$$ For an $RC$ circuit, the capacitor voltage is given by $$V_C=\dfrac{\varepsilon_0X_C}{\sqrt{R^2+X_C^2}}$$ where $X_C=1/\omega C$; $$V_C=\dfrac{\varepsilon_0}{\omega C}\dfrac{1}{\sqrt{R^2+\dfrac{1}{\omega^2 C^2}}}$$ $$V_C= \dfrac{\varepsilon_0}{\sqrt{\omega^2 C^2R^2+1}}$$ where $\omega=2\pi f$; $$V_C= \dfrac{\varepsilon_0}{\sqrt{ 4\pi^2 f^2 C^2R^2+1}}$$ Plug the known; $$\boxed{V_C= \dfrac{(10)}{\sqrt{ 4\pi^2 f^2 (10^{-6})^2(16)^2+1}}}$$ $\bullet$ At $f=1$ kHz, $$V_C= \dfrac{(10)}{\sqrt{ 4\pi^2 (1\times 10^3)^2 (10^{-6})^2(16)^2+1}}=\color{red}{\bf 9.95}\;\rm V$$ $\bullet$ At $f=3$ kHz, $$V_C= \dfrac{(10)}{\sqrt{ 4\pi^2 (3\times 10^3)^2 (10^{-6})^2(16)^2+1}}=\color{red}{\bf 9.57}\;\rm V$$ $\bullet$ At $f=10$ kHz, $$V_C= \dfrac{(10)}{\sqrt{ 4\pi^2 (10\times 10^3)^2 (10^{-6})^2(16)^2+1}}=\color{red}{\bf 7.05}\;\rm V$$ $\bullet$ At $f=30$ kHz, $$V_C= \dfrac{(10)}{\sqrt{ 4\pi^2 (30\times 10^3)^2 (10^{-6})^2(16)^2+1}}=\color{red}{\bf 3.15}\;\rm V$$ $\bullet$ At $f=100$ kHz, $$V_C= \dfrac{(10)}{\sqrt{ 4\pi^2 (100\times 10^3)^2 (10^{-6})^2(16)^2+1}}=\color{red}{\bf 0.99}\;\rm V$$ $$\color{blue}{\bf [b]}$$ Plugging the five points we got, to draw the $V_C$ versus $f$ graph. $\bullet$ $\rm (1\;kHz,9.95\;V)$ $\bullet$ $\rm (3\;kHz,9.57\;V)$ $\bullet$ $\rm (10\;kHz,7.05\;V)$ $\bullet$ $\rm (30\;kHz,3.15\;V)$ $\bullet$ $\rm (100\;kHz,0.99\;V)$
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