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 28 - The Electric Potential - Exercises and Problems - Page 834: 25

Answer

$\approx -1.6\;\rm kV$

Work Step by Step

We know that the net potential is the sum of the potentials due to each charge at the needed dot. Let's assume that $ q_1=1$ nC is the upper one, $ q_2=-2$ nC is the left one, and $q_3=-2$ nC is the right one. $$V_{net}=V_{\rm 4nC}+V_{\rm -2nC,right}+V_{\rm -2nC,left}$$ $$V_{net}=\dfrac{kq_1}{r_1}+\dfrac{kq_2}{r_2}+\dfrac{kq_3}{r_3}$$ $$V_{net}=k\left[\dfrac{ q_1}{r_1}+\dfrac{ q_2}{r_2}+\dfrac{ q_3}{r_3}\right]$$ It is obvious that the 3 charges form an equilateral triangle, and that the given dot is just at the center of this triangle. Hence $r_1=r_2=r_3=r$. $$V_{net}=k\left[\dfrac{ q_1}{r}+\dfrac{ q_2}{r}+\dfrac{ q_3}{r}\right]$$ $$V_{net}=\dfrac{k}{r}\left[q_1+ q_2+ q_3 \right]\tag 1$$ From the geometry of the right triangle, in the figure below, we can see that $\cos30^\circ=\dfrac{1.5}{r_1}$, and hence, $$r_1=\dfrac{1.5}{\cos30^\circ}=\bf \sqrt 3\;\rm cm$$ Plug the known into (1) $$V_{net}=\dfrac{(9\times 10^9)}{\sqrt{3}\times 10^{-2}}\left[1-2-2\right]\times 10^{-9}$$ $$V_{net}=\color{red}{\bf -1559}\;\rm V$$
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