Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.11 Application of Taylor Polynomials - 11.11 Exercises - Page 822: 36


$V \approx\dfrac{\pi k_e \sigma R^2}{d}$ for large $d$

Work Step by Step

Consider $\sqrt{d^2+R^2}=\sqrt{d^2(1+\dfrac{R^2}{d^2})}=d(1+\dfrac{R^2}{d^2})^{1/2}$ Now, need to apply the Binomial series. $d(1+\dfrac{R^2}{d^2})^{1/2}=d(1+\dfrac{R^2}{2d^2}+...)=d+\dfrac{R^2}{2d}+....$ The expression for potential is given as below: $V \approx 2\pi k_e \sigma (d+\dfrac{R^2}{2d}+......-d)$ This can be further rewritten as below: $V \approx 2\pi k_e \sigma (\dfrac{R^2}{2d})$ Hence, we get $V \approx\dfrac{\pi k_e \sigma R^2}{d}$ for large $d$
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