If electric charge did not exist, and protons and electrons were only bound together by gravitational forces to form hydrogen, derive the expressions for a_0 and E_n and compute the energy and frequency of the H_alpha line and limit of Balmer series.
E_n=mc^2 * alpha * 1 / (2n^2)
The Attempt at a Solution
If electric charge did not exist, then the balance of electron orbit and distance would change. Bohr's radius would just be zero i would imagine since it depends on electron charge. E_n would suffer the same fate... Would E_n just become mc^2? The radius would have to still be a number as the two do still have mass and gravity would affect them.
F=Gm1m2/r^2 is the force on either exerted by the other and
E=-Gm1m2/r is the gravitational potential energy
The energy of the electron E_n must be strictly dependent on this.
How do I "derive" an expression from E_n or a_0 when their original formulas must be completely nixed?
Is there supposed to be a way of expressing it with n levels?
Otherwise I propose the energy of the electron is -Gm1m2/r as there is no kinetic energy.