Suppose potential energy between electron and proton at separation r is given by U = K log r, where K is a constant. For such a hypothetical hydrogen atom, calculate radius of n th Bohr’s orbit and energy levels.
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Sol. If potential energy of a system of two particles, separated by a distance r is U, interaction force between the particles is F = – 
.
Since, potential energy of system of electron and proton in a hypothetical hydrogen atom is given by U = K log r, therefore, force between them, F = – 
= – 
Negative sign indicates force of attraction. This force of
attraction between electron and proton provides
centripetal force required for circular motion of
electron.
∴ 
= 
where m is mass of electron and v is its
speed, or mv 2 = K … (1)
According to Bohr’s postulates, angular momentum,
mvr = n 
… (2)
From equations (1) and (2),
r = 
total energy of the atom is E = potential energy U + kinetic energy 
of electron
E = K log r + 
K = 
(2 log r + 1) = 
(1 + log r 2 )
substituting value of r, E = 
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