Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 11 - Infinite Sequences and Series - 11.11 Exercises - Page 799: 33

Answer

$E$ is approximately proportional to $ \dfrac{1}{D^3}$ when $P$ is far away from the dipole.

Work Step by Step

Here, we have $E=\dfrac{q}{D^2}-\dfrac{q}{(D+d)^2}=\dfrac{q}{D^2}(1-\dfrac{1}{(1+\dfrac{d}{D})^2})$ This can be re-arranged as: $E=\dfrac{q}{D^2}-\dfrac{q}{(D+d)^2}=\dfrac{q}{D^2}(1-\dfrac{1}{(1+\dfrac{d}{D})^2})\approx \dfrac{q}{D^2}[1-(1-2(\dfrac{d}{D})+3(\dfrac{d}{D})^2-4(\dfrac{d}{D})^3)]$ $E\approx \dfrac{qd}{D^3}[2-3(\dfrac{d}{D})+4(\dfrac{d}{D})^2]$ We can see that when the point $P$ is far away from the dipole, then $\dfrac{d}{D}$ becomes very small, so this term can be ignored. Hence, $E\approx \dfrac{2qd}{D^3}$ Hence, it has been proved that $E$ is approximately proportional to $ \dfrac{1}{D^3}$ when $P$ is far away from the dipole.
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