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: 7

Answer

$\bf Less\;than.$

Work Step by Step

It is obvious, from the given graph, that the current leads the emf. This means that $\phi\lt 0$ where $\phi$ is given by $$\phi=\tan^{-1}\left[ \dfrac{X_{\rm L}-X_{\rm C}}{R} \right]$$ So, $$ \tan^{-1}\left[ \dfrac{X_{\rm L}-X_{\rm C}}{R} \right]\lt 0$$ which means that $$ X_{\rm L}-X_{\rm C} \lt 0\tag 1$$ Recalling that: $X_{\rm C}=1/\omega C$, and $X_{\rm L}=\omega L$, Plug those into (1), $$ \omega L-\dfrac{1}{\omega C} \lt 0 $$ $$\dfrac{ \omega^2 LC-1}{\omega C} \lt 0 $$ $$ \omega^2 LC-1 \lt 0 $$ $$ \omega^2 LC \lt 1 $$ $$ \omega^2 \lt \dfrac{1}{LC} $$ $$ \omega \lt \sqrt{\dfrac{1}{LC} }$$ where $\omega_0=1/\sqrt{LC}$, so $$\boxed{ \omega \lt \omega_0 }$$
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