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(A) The potential energy $U$ is given by $U = eV = eV_0 \ln(r/r_0)$.The force $F$ is given by $F = -dU/dr = -d/dr(eV_0 \ln(r/r_0)) = -eV_0/r$. The mag

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$r_n \propto n$

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(A) The potential energy $U$ is given by $U = eV = eV_0 \ln(r/r_0)$.The force $F$ is given by $F = -dU/dr = -d/dr(eV_0 \ln(r/r_0)) = -eV_0/r$. The magnitude of the force is $F = eV_0/r$.This force provides the necessary centripetal force for circular motion: $mv^2/r = eV_0/r$.Simplifying this,we get $mv^2 = eV_0$,which implies $v = \sqrt{eV_0/m}$. Note that $v$ is independent of $r$ and $n$.According to Bohr's quantization condition,the angular momentum is $mvr = nh/(2\pi)$.Substituting $v = \sqrt{eV_0/m}$ into the quantization condition,we get $m(\sqrt{eV_0/m})r_n = nh/(2\pi)$.Solving for $r_n$,we get $r_n = (nh / (2\pi)) \cdot \sqrt{1/(meV_0)}$.Since $h, m, e, V_0$ are constants,we conclude that $r_n \propto n$.

The electric potential between a proton and an electron is given by $V = V_0 \ln(r/r_0)$,where $r_0$ is a constant. Assuming Bohr's model to be applicable,find the variation of $r_n$ with $n$,where $n$ is the principal quantum number.

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