Using Rutherford model of atom , derive the expression for the total energy of the electron in hydrogen atom. What is the significance of total negative energy possessed by the electron ?

Explanation

Coulomb attractive force between the nucleus (proton ) of charge +e and the electron of charge -e is
`F=1/(4piin_0)*(e^2)/(r^2)=(k)/(r^2) ("say")`
We assume the proton to be too heavy compared to the electron . Then the proton remains stationary when the electron revolves in a circular orbit of radius r with velocity v

The force F provides the centripetal force `(mv^2)/(r)` for this revolution (m=electron mass).
`therefore (mv^2)/(r) =k/(r^2)`
The kinetic energy ,
`E_k=1/2 mv62=k/(2r)`
The potential energy = work done to bring the electron from an infinite distance , i.e.,
`E_p=int_(oo)^r k/(r^2)dr=k[-1/r]_(oo)^r=-k/r`
`therefore` Total energy,
`E=E_k+E_p=k/(2r)-k/r=-k/(2r)=-1/(4piin_0)*(e^2)/(2r)`
This negative of E suggest that the electron is bound in hydrogen atom. An equal amount of positive energy is to be supplied from outside to make the total energy of the electron to be zero. Then the electron will be free from the atom.