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

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.