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 29 - Potential and Field - Exercises and Problems - Page 864: 44

Answer

$40\;\rm V$, $\approx 27^\circ$

Work Step by Step

We know that the electric field is given by $$\vec E=-\left[ \dfrac{\partial V}{\partial x}\;\hat i+\dfrac{\partial V}{\partial y}\;\hat j\right]$$ where $V=\dfrac{200}{\sqrt{x^2+y^2}}=200(x^2+y^2)^{-\frac{1}{2}}$ $$\vec E=-\left[ \dfrac{\partial [200(x^2+y^2)^{-\frac{1}{2}})]}{\partial x}\;\hat i+\dfrac{\partial [200(x^2+y^2)^{-\frac{1}{2}}]}{\partial y}\;\hat j\right]$$ $$\vec E=-\left[ 200 (-\frac{1}{2}) (2x)(x^2+y^2)^{-3/2}\;\hat i+200 (-\frac{1}{2}) (2y)(2^2+1^2)^{-3/2}\;\hat j\right]$$ $$\boxed{\vec E= 200x(x^2+y^2)^{-3/2}\;\hat i+200y(x^2+y^2)^{-3/2}\;\hat j}$$ At $(x,y)=(2\;{\rm m}, 1\;{\rm m})$, $$\vec E= 200(2)(2^2+1^2)^{-3/2}\;\hat i+200(1)(x^2+y^2)^{-3/2}\;\hat j$$ $$\vec E= ( {\bf 35.777}\;\hat i+{\bf 17.888}\;\hat j )\;\rm V/m$$ So its magnitude is given by $$E=\sqrt{E_x^2+E_y^2}=\sqrt{(35.777)^2+(17.888)^2}$$ $$E=\color{red}{\bf 40}\;\rm V/m$$ Since the $E_x$ is positive and $E_y$ is also positive, this is the first quadrant. Thus, its direction, which is counterclockwise from the positive $x$-axis, is given by $$\theta= \tan^{-1}\left[ \dfrac{|E_y|}{|E_x|}\right]$$ $$\theta= \tan^{-1}\left[ \dfrac{|17.888|}{|35.777|}\right]$$ $$\theta= \color{red}{\bf 26.6}^{\circ}\tag {CCW from $+x$-axis}$$
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