Chapter 21 Problem 86

Problem

An electron moves in a circle of radius $r$ around a very long uniformly charged wire in a vacuum chamber. The charge density on the wire is $\lambda =0.14\mu {\textrm {C}}/{\textrm {m}}$.

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

(b) What is the speed of the electron?

solution

(a)

Has been done in the lecture.

$E={\frac {1}{2\pi \epsilon _{0}}}{\frac {\lambda }{r}}$ which is radially away from the wire at all points.

(b)

The force on the electron will point radially in, producing a centripetal acceleration.

$\left|{\vec {F}}\right|=\left|q{\vec {E}}\right|={\frac {e}{2\pi \epsilon _{0}}}{\frac {\lambda }{r}}$

Using the centripetal term (from last year)

$\left|{\vec {F}}\right|={\frac {mv^{2}}{r}}$

$v={\sqrt {2{\frac {1}{4\pi \epsilon _{0}}}{\frac {e\lambda }{m}}}}={\sqrt {2(8.99\times 10^{9}{\textrm {N}}\cdot {\textrm {m}}^{2}/{\textrm {C}}^{2}{\frac {(1.6\times 10^{-19}{\textrm {C}})(0.14\times ^{-6}{\textrm {C}}/{\textrm {m}})}{9.11\times 10^{-31}{\textrm {kg}}}}}}=2.1\times 10^{7}{\textrm {m}}/{\textrm {s}}$

independent of r!