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 27 - Gauss's Law - Exercises and Problems - Page 807: 31

Answer

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Work Step by Step

$$\color{blue}{\bf [a]}$$ We know that the electric flux through a surface is given by $$\Phi=\vec E\cdot \vec A=EA\cos\theta$$ The area of the base of the tetrahedron, as we see in the figure below, is given by $A_{\rm base}=0.5L h$ where $h=\sqrt{L^2-(0.5L)^2}=\sqrt{3} L/2$. Hence, $A=\frac{\sqrt3}{4}L^2$ Thus, $$\Phi_{\rm base} =\frac{\sqrt3}{4}EL^2\cos180^\circ$$ Plug the known; $$\Phi_{\rm base} =-\frac{\sqrt3}{4}(200)(0.2)^2 $$ $$\Phi_{\rm base} =\color{red}{\bf -3.46}\;\rm N\cdot m^2/C$$ $$\color{blue}{\bf [b]}$$ Since the tetrahedron has three equilateral triangle sides. and since the electric field is uniform and pointing upward, the three sides will get the same amount of flux. Hence, $$\Phi_{\rm tetrahedron }=\Phi_{\rm base}+3\Phi_{\rm sides}$$ Thus, $$\Phi_{\rm sides}=\dfrac{\Phi_{\rm tetrahedron }-\Phi_{\rm base}}{3}=\dfrac{0-\Phi_{\rm base}}{3}$$ where $\Phi_{\rm tetrahedron }=0$ since it has no inner charge. Plugging the known; $$\Phi_{\rm sides}=\dfrac{ -( -3.46)}{3}=\color{red}{\bf 1.15}\;\rm N\cdot m^2/C$$ Each side will be affected by this amount of electric flux.
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