Question

In Bohr's model of hydrogen atom, the centripetal force is provided by the Coulomb attraction between the proton and the electron. If $a_0$ is the radius of the ground state orbit, m is the mass and e the charge of an electron and ${ϵ}_0$ is the vacuum permittivity, the speed of the electron is

• A. zero
• B. $\frac{e}{\sqrt{{ϵ}_0{a}_{0}m}}$
• C. $\frac{e}{\sqrt{4\pi m{ϵ}_0{a}_0}}$
• D. $\frac{\sqrt{4\pi {ϵ}_0{a}_{0}m}}{e}$

Answer 1

The correct option is C $\frac{e}{\sqrt{4\pi m{ϵ}_0{a}_0}}$

Centripetal force, $F=\frac{m{v}^2}{a_0}$
Coulomb's force between the electron and the proton, $F^{\prime }=\frac{1}{4\pi {ϵ}_0}\frac{e^2}{a_0^2}$
Now, $F=F^{\prime }\Rightarrow \frac{m{v}^2}{a_0}=\frac{1}{4\pi {ϵ}_0}\frac{e^2}{a_0^2}\Rightarrow v=\frac{e}{\sqrt{4\pi m{ϵ}_0{a}_0}}$
Hence, the correct choice is (c).