Question

A $2.0-mm$-diameter glass sphere has a charge of $+1.0nC$. What speed does an electron need to orbit the sphere $1.0mm$ above the surface?

Short answer

The electron needs to orbit the sphere at a speed of $\mathbf{3}\mathbf{.}\mathbf{97}\mathbf{}\mathit{x}\mathbf{}{\mathbf{10}}^{\mathbf{7}}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$

Step-by-step solution

Given information and formula 

Given :

Diameter of glass sphere : $2.0-mm$

Charge on glass sphere : $+1.0nC$

Distance of the sphere from the surface :$1.0mm$

Theory used :

Electrostatic force on a sphere concentric on the center generates the electron centripetal force. That is :

$F_e=F_c$

Calculating the orbital speed

Electrostatic force on a sphere concentric on the center generates the electron centripetal force.

$$\begin{gathered} F_e=F_c \\ \Rightarrow \frac{1}{4\pi {\epsilon }_0}\times \frac{Q_e}{r^2}=\frac{mv²}{r} \\ \Rightarrow \frac{9\times {10}^9\times {10}^{-9}\times 1.6\times {10}^{-19}}{{10}^{-3}}=9.11\times {10}^{-31}v² \\ \Rightarrow v^2=\frac{1.44x{10}^{-15}}{9.11x{10}^{-31}} \\ \Rightarrow v=\sqrt{1.15068x{10}^{-47}} \\ =\boxed{\mathbf{3}\mathbf{.}\mathbf{97}\mathbf{}\mathit{x}\mathbf{}{\mathbf{10}}^{\mathbf{7}}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}} \end{gathered}$$