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(A) The potential energy of the system is $U = eV = eV_0 \ln \left( \frac{r}{r_0} \right)$.The force $F$ is given by $F = -\frac{dU}{dr} = -\frac{eV_0

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$r_n \propto n$

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(A) The potential energy of the system is $U = eV = eV_0 \ln \left( \frac{r}{r_0} \right)$.The force $F$ is given by $F = -\frac{dU}{dr} = -\frac{eV_0}{r}$.This force provides the necessary centripetal force for circular motion:$\frac{mv^2}{r} = \frac{eV_0}{r} \implies v = \sqrt{\frac{eV_0}{m}}$.According to Bohr's quantization condition,the angular momentum is $mvr = n \frac{h}{2\pi}$.Substituting the expression for $v$:$mr \left( \sqrt{\frac{eV_0}{m}} \right) = n \frac{h}{2\pi}$.Since $m, e, V_0, h$ are constants,we have $r \propto n$.Therefore,$r_n \propto n$.

The potential energy of a proton and an electron in a hydrogen atom is given by $V = V_0 \ln \left( \frac{r}{r_0} \right)$,where $r_0$ is a constant. Assuming the system follows the Bohr model,find the relationship between the radius $r_n$ and the principal quantum number $n$.

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