Here is the essence of Bohr's 1913 paper which appeared in Volume 26 of the Philosophical Magazine and was entitled "On the Constitution of Atoms and Molecules". He started by assuming a circular orbit for electrons around the nucleus. The electron charge is "e" and the nucleus has "Z" positive charges, each of magnitude "e". "m" is the electron mass, "v" is its velocity, and "r" is the radius of its orbit.
The total energy "E" of such a system is the sum of its kinetic and potential energy. This sum must be a constant for the atom to be stable. The kinetic energy comes simply from its elementary definition and the potential energy is just Coulomb's Law.
Furthermore, for this system to be stable, it is assumed that the attractive Coulomb force of the charged particles is balanced by the centripetal force of the rotating electron. These basic definitions give us this relation.
By combining these two expressions, we arrive at the following expression for the energy of the bound system. The bound system energy is negative while if the electron is freed from the nucleus, r=&inf;, the energy is 0.
The circumference of the orbit is, of course, 2πr, and when travelling at velocity v, the frequency of the orbit is simply
We can use the above relations to eliminate r and v and obtain an expression for the frequency as
Now here is another big assumption. Scientists liked to associate the radiation from an orbiting particle with its mechanical orbital frequency. Here is an electron orbiting with the frequency given above. Bohr assumed that it was stable but that energy would be given up when it was taken out of its orbit to infinity. There its frequency would be zero. He suggested that the frequency of the emitted light would be the average of the mechanical frequency before and after the transition, which is obviously just f/2.