derivations of the total energy of an electron in an orbit  according to bohr’s atomic model

if me is the mass of an electron revolving around an orbit with radius rn,

the centripetal force on the electron  must be balanced  by the coulomb’s force  between the electron and the nucleus

. thus

where

note substitute d with π and E with epsilon ℇ

thus  

total energy = kinetic energy  + potential energy

but  

thus      substituting    into the equation below we have

 .

THE POTENTIAL ENERGY  

now is to put the value  the above equation

according to bohr’s 2nd postulate , momentum of electron is equal to nh/2π

putting the value  of v in the following equation we have

substituting   into the equation below we get

where

This is the total energy of an electron in n orbit revolving around the nucleus.

note substitute d with π and E with epsilon ℇ