An electron is rotating around an infinite positive linear charge in a circle of radius of 0.1 m....

Question:

An electron is rotating around an infinite positive linear charge in a circle of radius of 0.1 m. If linear charge density is 1 micro coulomb per meter, velocity of the electron in m/s is _____.

Electric Field due To Line of Charge

The electric field due to a uniformly charged infinite line of the charge depends upon the linear charge density of the line of charge and the distance from the center of the line of charge. Mathematically

{eq}\begin{align} E = \frac{2 k \lambda }{r} \end{align} {/eq}

Where r is the distance from the center of the line and point of observation and {eq}\lambda {/eq} is the linear charge density.

Data Given

• Radius of the circular path {eq}r = 0.10 \ \rm m {/eq}
• Linear charge density of the infinite linear charge {eq}\lambda = 10^{-6} \ \rm C/m {/eq}

In this the case, the necessary centripetal charge is provided by the electrostatic force between the electron an infinite linear charge, i.e

{eq}\begin{align} \frac{mv^2}{r} = qE \\ \frac{mv^2}{r} = \frac{2 k \lambda q}{r} \\ mv^2 = 2 k \lambda q \\ v = \sqrt{ \frac{ 2 k \lambda q}{m}} \\ v = \sqrt{ \frac{ 2 \times 9 \times 10^9 \ \rm N.m^2C^{-2} \times 10^{-6} \ \rm C/m \times 1.6 \times 10^{-19} \ \rm C}{9.11 \times 10^{-31} \ \rm kg}} \\ \color{blue}{\boxed{ \ v = 5.62 \times 10^7 \ \rm m/s \ }} \end{align} {/eq}