University Physics with Modern Physics (14th Edition)

Published by Pearson
ISBN 10: 0321973615
ISBN 13: 978-0-32197-361-0

Chapter 24 - Capacitance and Dielectrics - Problems - Exercises - Page 811: 24.43

Answer

(a) $E = \dfrac{Q}{\epsilon_o K A} $ (b) $V = \dfrac{Q}{\epsilon_o K A} d$ (c) $C = \dfrac{\epsilon_o K A}{d}$

Work Step by Step

(a) We integrate equation 24.23 for $A$ to determine the electric field by \begin{gather*} \oint K \vec{E} \cdot d\vec{A} = \dfrac{Q}{\epsilon_o} \\ K E A = \dfrac{Q}{\epsilon_o} \\ \boxed{E = \dfrac{Q}{\epsilon_o K A} } \end{gather*} (b) We use the expression of the electric field to determine the potential $V$ by $$V = E d = \boxed{\dfrac{Q}{\epsilon_o K A} d}$$ (c) The capacitance depends on the charge and the potential and it is given by \begin{gather*} C = \dfrac{Q}{V}\\ C = \dfrac{Q}{ \dfrac{Q}{\epsilon_o K A} d }\\ \boxed{C = \dfrac{\epsilon_o K A}{d}} \end{gather*} This is the same for equation 24.2 but after inserting a dielectric material
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