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 26 - The Electric Field - Exercises and Problems - Page 777: 46

Answer

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Work Step by Step

$\bullet$ At point 1: The direction of the electric field of the left sheet is toward the sheet itself which means toward the right while the direction of the electric field of the right sheet is away from the sheet which means toward the left. Thus, the net electric field at point 1 is given by $$E_{net}=E_{left}-E_{right}$$ where $E_{sheet}=\eta/2\epsilon_0$, so $$E_{net}=\dfrac{ \eta_{left}}{2\epsilon_0}\;\hat i-\dfrac{ \eta_{right}}{2\epsilon_0}\;\hat i$$ $$E_{net}=\dfrac{|-\eta_0|}{2\epsilon_0}\;\hat i-\dfrac{|3\eta_0|}{2\epsilon_0}\;\hat i$$ $$\boxed{E_{net}= -\dfrac{ \eta_0 }{ \epsilon_0}\;\hat i}$$ $\bullet\bullet$ At point 2: The direction of the electric field of the left sheet is toward the sheet itself which means toward the left while the direction of the electric field of the right sheet is away from the sheet which means toward the left as well. Thus, the net electric field at point 1 is given by $$E_{net}=-(E_{left}+E_{right})$$ $$E_{net}=-\left[\dfrac{ \eta_{left}}{2\epsilon_0}\;\hat i+\dfrac{ \eta_{right}}{2\epsilon_0}\;\hat i \right]$$ $$\boxed{E_{net}= -\dfrac{ 2\eta_0 }{ \epsilon_0}\;\hat i}$$ $\bullet\bullet\bullet$ At point 3: The direction of the electric field of the left sheet is toward the sheet itself which means toward the left while the direction of the electric field of the right sheet is away from the sheet which means toward the right. Thus, the net electric field at point 1 is given by $$E_{net}=E_{right}-E_{left}$$ where $E_{sheet}=\eta/2\epsilon_0$, so $$E_{net}=\dfrac{ \eta_{right}}{2\epsilon_0}\;\hat i-\dfrac{ \eta_{left}}{2\epsilon_0}\;\hat i $$ $$E_{net}=\dfrac{|3\eta_0|}{2\epsilon_0}\;\hat i-\dfrac{|-\eta_0|}{2\epsilon_0}\;\hat i$$ $$\boxed{E_{net}= \dfrac{ \eta_0 }{ \epsilon_0}\;\hat i}$$
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