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 25 - Electric Charges and Forces - Exercises and Problems - Page 748: 60

Answer

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Work Step by Step

$\Rightarrow$ At point (1), the electric field is directly upward since the charge is positive so it is always away from the charge. And its magnitude is given by $$E_1=\dfrac{kq}{r_1^2}$$ Plugging the known (see the given graph), $$E_1=\dfrac{(8.99\times 10^{9})(10\times 10^{-9})}{(0.03)^2}\;\hat j=(\color{red}{\bf 1.0\times 10^5}\;{\rm N/C})\hat j$$ $\Rightarrow$ At point (2), the electric field is given by $$E_2=\dfrac{kq}{r_2^2}\cos\theta\;\hat i+ \dfrac{kq}{r_2^2}\sin\theta\;\hat j$$ where, from the geometry of the given figure, $\cos\theta= \dfrac{0.04}{r_2}$, $\sin\theta=\dfrac{0.03}{r_2}$, and $r_2=\sqrt{0.03^2+0.04^2}$. Thus, $$E_2=\dfrac{kq}{r_2^2} \cdot\dfrac{0.04}{r_2}\;\hat i+ \dfrac{kq}{r_2^2} \dfrac{0.03}{r_2}\;\hat j$$ $$E_2=\dfrac{0.04kq}{r_2^3} \;\hat i+ \dfrac{0.03kq}{r_2^3} \;\hat j$$ $$E_2=\dfrac{kq}{r_2^3} \left[0.04 \;\hat i+0.03 \;\hat j\right]$$ Plugging the known (see the given graph), $$E_2=\dfrac{(8.99\times 10^{9})(10\times 10^{-9})}{(\sqrt{0.03^2+0.04^2})^3} \left[0.04 \;\hat i+0.03 \;\hat j\right]$$ $$E_2 =(\color{red}{\bf 2.9\times 10^4}\;{\rm N/C})\hat i+(\color{red}{\bf 2.2\times 10^4}\;{\rm N/C})\hat j$$ $\Rightarrow$ At point (3), the electric field is directly eastward since the charge is positive so it is always away from the charge. And its magnitude is given by $$E_3=\dfrac{kq}{r_3^2}$$ Plugging the known (see the given graph), $$E_3=\dfrac{(8.99\times 10^{9})(10\times 10^{-9})}{(0.04)^2}\;\hat i=(\color{red}{\bf 5.6\times 10^4}\;{\rm N/C})\hat i$$
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