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 - Conceptual Questions - Page 1051: 8

Answer

$\bf Lags.$

Work Step by Step

This is the opposite of the previous problem in which we found that $\phi\lt 0$ and this gave us $\omega\lt \omega_0$. We are given that $$\omega\gt \omega_0$$ where $\omega_0=1/\sqrt{LC}$, so $$\omega\gt \sqrt{\dfrac{1}{LC} }$$ $$\omega^2\gt \dfrac{1}{LC} \tag 1 $$ Recalling that: $X_{\rm C}=1/\omega C$, so $C=1/\omega X_{\rm C}$, and $X_{\rm L}=\omega L$, so $L=X_{\rm L}/\omega$ Plug into (1), $$\omega^2\gt \dfrac{1}{LC} $$ $$\omega^2\gt \dfrac{1}{(X_{\rm L}/\omega)(1/\omega X_{\rm C})} $$ $$ \color{red}{\bf\not} \omega^2\gt \dfrac{ \color{red}{\bf\not} \omega^2 X_{\rm C}}{ X_{\rm L} } $$ $$1\gt \dfrac{ X_{\rm C}}{ X_{\rm L} } $$ Therefore, $$ X_{\rm L}\gt X_{\rm C}$$ $$ X_{\rm L}-X_{\rm C}\gt 1 $$ which means that $\boxed{\phi\gt 0}$ where $$\phi=\tan^{-1}\left[ \dfrac{X_{\rm L}-X_{\rm C}}{R} \right]$$ This means that the current lags the emf.
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