In the Bohr 's model of hydrogen-like atom the force between the nucleus and the electron is modified as \(F = \frac{{{e^2}}}{{4\pi {\varepsilon _0}}}\left( {\frac{1}{{{r^2}}} + \frac{\beta }{{{r^3}}}} \right)\), where \(\beta \) is a constant. For this atom, the radius of the \(n^{th}\) orbit in terms of the Bohr radius \(\left( {{a_0} = \,\frac{{{\varepsilon _0}{h^2}}}{{m\pi {e^2}}}} \right)\) is

and \(\mathrm{mvr}=\frac{\mathrm{nh}}{2 \pi} \Rightarrow \mathrm{v}=\frac{\mathrm{nh}}{2 \pi \mathrm{m} \mathrm{r}}\)

\(\therefore m{\left( {\frac{{{\text{nh}}}}{{2\pi {\text{mr}}}}} \right)^2} \times \frac{1}{{\text{r}}}\) \( = \frac{{{{\text{e}}^2}}}{{4\pi {_0}}}\left( {\frac{1}{{{r^2}}} + \frac{{\text{B}}}{{{{\text{r}}^3}}}} \right)\)

or, \(\frac{1}{r^{2}}+\frac{B}{r^{3}}=\frac{m n^{2} h^{2} 4 \pi \epsilon_{0}}{4 \pi^{2} m^{2} e^{2} r^{3}}\)

or, \(\frac{a_{0} n^{2}}{r^{3}}=\frac{1}{r^{2}}+\frac{B}{r^{3}}\)

\(\left(\because a_{0}=\frac{\epsilon_{0} h^{2}}{m \pi e^{2}} \text { Given }\right)\)

\(\therefore r=a_{0} n^{2}-B\)