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Soppose potential energy between electronand proton at seperation `r` is given by `U = klog r, where k` is a constant. For such a hypothetical hydrogen atom , calculate the radins of nth Bohr and its energy level

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For a conservative force field, `F = - (dU)/(dr) = - (k)/( r)`
This force `F = -k//r` provides the centripetal force for the circular motion of electron
So `(m nu^(2))/( r) = (k)/( r) implies m^(2) nu^(2) = mk implies m nu = sqrtmk` …(i)
Applying bohr's quantization rule, `m nu r = (nh)/(2 pi)` ...(ii)
From Eqs. (i) and (ii) , we get `r = (nh)/(2 pi sqrtmk)`
From Eqs. (i), `KE` of electron `= (1)/(2) m nu^(2) = (1)/(2) k`
Total energy of electron `= K E + P E = (1)/(2) k + k log r`
`= (k)/(2) [1 + log (n^(2) h^(2))/(4 pi ^(2) mk)]`
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