View source for Chapter 21 Problem 86

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==Problem==
[[File:Chapter21-problem86q.png|200px|right|line charge and orbiting electron]]
An electron moves in a circle of radius <math>r</math> around a very long uniformly charged wire in a vacuum chamber. The charge density on the wire is <math>\lambda = 0.14 \mu\textrm{C}/\textrm{m}</math>. 

(a) What is the electric field at the electron (magnitude and direction in terms of <math>r</math> and <math>\lambda</math> )? 

(b) What is the speed of the electron?

==solution==
===(a)===
Has been done in the lecture. 

<math>E=\frac{1}{2\pi\epsilon_0}\frac{\lambda}{r}</math> which is radially away from the wire at all points. 

===(b)===
The force on the electron will point radially in, producing a centripetal acceleration.

<math>\left|\vec{F}\right|=\left|q\vec{E}\right|=\frac{e}{2\pi\epsilon_0}\frac{\lambda}{r}</math>

Using the centripetal term (from last year)
 
<math>\left|\vec{F}\right|=\frac{mv^2}{r}</math> 

<math>v=\sqrt{2\frac{1}{4\pi\epsilon_0}\frac{e\lambda}{m}}=\sqrt{2(8.99\times 10^9 \textrm{N}\cdot\textrm{m}^2/\textrm{C}^2\frac{(1.6\times10^{-19}\textrm{C})(0.14\times^{-6}\textrm{C}/\textrm{m})} {9.11\times10^{-31}\textrm{kg}}}=2.1\times10^7 \textrm{m}/\textrm{s}</math>

independent of r!