Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)

Published by Pearson
ISBN 10: 0133942651
ISBN 13: 978-0-13394-265-1

Chapter 23 - The Electric Field - Stop to Think 23.2 - Page 636: 1

Answer

$\eta_a = \eta_b = \eta_c$

Work Step by Step

The wording of this question is key. Note that the question says the piece of plastic is$\quad UNIFORMLY$ charged. This means that there's an equal amount of charge distributed throughout the surface of the plastic. While the surface charge density is defined in terms of charge per area, we know that even though when the plastic is broken up into two pieces and piece $b$ is bigger and has more charge on it, that its surface charge density is EQUAL to that of piece $c$ because piece $b$ also has a larger surface area. Mathematically, even though $Q_b > Q_c$ and $A_b > A_c$, that $\displaystyle \frac{Q_b}{A_b} = \frac{Q_c}{A_c}$. A charge uniformly distributed means that surface charge $Q$ and surface area $A$ are directly proportional to one another (linearly related); $Q \propto A$ . As for piece $a$, again, even though it's bigger than both pieces $b$ and $c$, its surface charge density is equal to the other pieces because its surface charge and surface area are directly proportional. $\therefore \eta_a = \eta_b = \eta_c$
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