## Derivation of the Energy of the Bohr model of a hydrogen atom

** Problem: Derive**

**Solution**

We start with the three equations below,

Equation 1 is an expression of the fact that the total energy E of an electron is the sum of its kinetic energy and its potential energy . Equation 2 is the fundamental equation of Newtonian dynamics, is the Coulomb force exerted on the electron and is the acceleration of its uniform circular motion. Lastly, equation 3 expresses the quantization condition introduced by Bhor to explain the existence of discrete energy levels.

From equation 2,

Substituting this to equation 1,

Now we solve for from equation 3 then substitute it to the E above.

From equation 2, we solve for v,

Finally we substitute v to E,

recall

Where we have the Ionization Energy

Thus,