Q. If we assume only gravitational attraction between proton and electron in hydrogen atom and the Bohr’s quantization rule to be followed, then the expression for the ground state energy of the atom will be (the mass of proton is M and that of electron is m)

(a) G^{2}M^{2}m^{2}/h^{2}

(b) −2π^{2}G^{2}M^{2}m^{3}/h^{2}

(c) −2π^{2}GM^{2}m^{3}/h^{2}

(d) None of these

**Ans: (b)**

Sol: As , mv^{2}/r = GMm/r^{2}

Hence , v = √(GM/r)

A/c to Bohr’s postulate

mvr = nh/2π

m {√(GM/r)}r = nh/2π

Squaring ,

m^{2}(GM/r)r^{2} = (nh/2π)^{2}

For ground state putting n = 1

r = (h/2π)^{2} × (1/GMm^{2}) …..(i)

Ground State Energy ,

E = 1/2(mv^{2}) + (-GMm/r)

= (GMm/2r) -GMm/r (Since , v=√(GM/r) )

= -GMm/2r

On putting the value of r from eqn(i)

E = −2π^{2}G^{2}M^{2}m^{3}/h^{2}