How Can the Bohr Radius Help Calculate Energy Levels in Hydrogen?

[mIKEjONES]

I'm currently reading a book which says that by knowing by the Bohr radius, in this example that of hydrogren, the energy of each orbit level can be calculated by deriving the appropriate Rydberg formula constant using Coulomb's law and Newtonian mechanics.

Following their description I tried deriving the energy possessed by each orbit level for hydrogen, my final solution should then be E = -\frac{13.6eV}{n^{2}} (E here is energy expressed in eV)
knowing the Bohr radius of hydrogen, the radius in terms of the energy level is
r=n^{2} a_0, where a_0 = 5.29*10^{-11} m .

I then substituted r into Coulomb's law
F = {1 \over 4\pi\varepsilon_0}\frac{q_1 * q_2}{r^2}= 9*10^9 mF^{-1} * {1.60217646*10^{-19}C * -1.60217646*10^{-19}C \over \left(n^{2} * 5.29*10^{-11} m)^2

After not really having accomplished much, I don't know how to proceed. Any suggestions, explanations are welcome.

Thank you :)

 

[0xDEADBEEF]

The Coulomb Force field is conservative (no curl no dependence on t) thus there is a potential V(r) \propto - \frac{1}{r} that you receive from integration. The Energy of an electron is the energy it needs to escape from the nucleus. If I were you I would calculate the Coulomb potential, plug in the radius which should give you a value for V(r). But Energy is potential plus kinetic energy (E = T + V) so you calculate the kinetic energy T from the speed of the electron. When the electron has escaped the energy is zero T+V=0, by definition of our integration constant.
Have fun!