Question

Derive an expression for potential and kinetic energy of an electron in any orbit of a hydrogen atom according to Bohr's atomic model.

Answer 1

Let an electron revolves with a constant speed $v$ around the nucleus in a stationary orbit of radius $r$. The electrostatic force acting on the electron is balanced by the centrifugal force i.e. $F_c=F_e$

$\therefore$ $\frac{m{v}^2}{r}=\frac{k{e}^2}{r^2}$ where $k=\frac{1}{4\pi {ϵ}_o}$

OR $m{v}^2=\frac{k{e}^2}{r}$ ......(1)

Thus kinetic energy of electron $K=\frac{1}{2}m{v}^2=\frac{k{e}^2}{2r}$ ......(2)

Potential energy of electron $U=\frac{-k{e}^2}{r}$

$⟹$ $U=-2K$ (using (2))