Question

Suppose the potential energy between electron and proton at a distance $r$ is given by $U=-ke$ in (r), where $k$ is a positive constant. Use Bohr's theory to obtain the energy of $n^{th}$ energy level for such an atom.

Answer 1

Since H atom is a bounded system U cannot be positive

$U=\frac{{-ke}^{2}2}{{3r}^3}$

Electrostatic force of attraction at a distance r will be

$F=\frac{-dU}{dr}=\frac{{-ke}^2}{r^4}$

$\frac{{mv}^2}{r}=\frac{{-ke}^2}{r^4}$

${mv}^2=\frac{{ke}^2}{r^3}$

$\frac{1}{2}{mv}^2=\frac{{ke}^2}{{2r}^3}$

$KE=\frac{{ke}^2}{{2r}^3}$

$E=KE+U$

$=ke\frac{2}{{2r}^3}+(-ke\frac{2}{{3r}^3})$

$=\frac{{ke}^2}{{6r}^3}$