Physics: Principles with Applications (7th Edition)

Published by Pearson
ISBN 10: 0-32162-592-7
ISBN 13: 978-0-32162-592-2

Chapter 21 - Electromagnetic Induction and Faraday's Law - Problems - Page 623: 72

Answer

$\approx 43\;\rm loops$

Work Step by Step

First of all, we need to find $L$, so we can find the number of loops $N$. We know that the frequency of a resonant circuit is given by $$f_0=\dfrac{1}{2\pi }\sqrt{\dfrac{1}{LC}}$$ Solving for $L$; $$(2\pi f_0 )^2= \dfrac{1}{LC} $$ $$L = \dfrac{1}{(2\pi f_0 )^2 C}\tag 1 $$ We also know that $$L=\dfrac{\mu_0N^2A}{l}$$ Plugging into (1); $$\dfrac{\mu_0N^2A}{l_{solenoid}} = \dfrac{1}{(2\pi f_0 )^2 C} $$ Noting that the length of the air-core solenoid conductor is given by the number of loops times the diameter of the wire. so $l_{solenoid}= ND_{wire}$ Thus, $$\dfrac{\mu_0N^2A}{ND_{wire}} = \dfrac{1}{ 4\pi^2 f_0^2 C} $$ The cross-sectional area of the solenoid is a circle, so its area is given by $A=\pi r^2$ where $r$ is the radius of one loop which is the radius of the solenoid itself. $$\dfrac{\mu_0N \pi r_{loop}^2 }{ D_{wire}} = \dfrac{1}{4\pi^2 f_0^2 C} $$ Solving for $N$; $$ N = \dfrac{D_{wire}}{4\mu_0\pi^3 f_0^2 r_{loop}^2 C} \tag 2$$ Recall that the number of loops is also given by the length of the wire divided by the circumference of one loop; $N=l_{wire}/2\pi r_{loop}$; thus $$r_{loop}=\dfrac{l_{wire}}{2\pi N}$$ Plugging into (2); $$ N = \dfrac{D_{wire}}{4\mu_0\pi^3 f_0^2\left(\dfrac{l_{wire}}{2\pi N}\right)^2 C} = \dfrac{4\pi ^2 N^2D_{wire}}{4\mu_0\pi^3 f_0^2l_{wire}^2 C} $$ $$ 1 = \dfrac{ N D_{wire}}{ \mu_0\pi f_0^2l_{wire}^2 C} $$ Therefore, $$N= \dfrac{ \mu_0\pi f_0^2l_{wire}^2 C}{ D_{wire}} $$ Plugging the known; $$N=\dfrac{ 4\pi^2\cdot 10^{-7}\cdot (18000)^2\cdot (12)^2\cdot 260\times10^{-9} }{ 1.1\times10^{-3}}=43.5\;\rm loop$$ $$N\approx \color{red}{\bf 43}\;\rm loop$$
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