The radius of the $n$th Bohr orbit in a hydrogen-like atom of nuclear charge $Z$ scales as
A$r_n \propto \tfrac{n}{Z^2}$
B$r_n \propto nZ$
C$r_n \propto \tfrac{n^2}{Z}$
D$r_n \propto \tfrac{Z}{n^2}$
Answer & Solution
Correct answer: C. $r_n \propto \tfrac{n^2}{Z}$
Equating the Coulomb force $\tfrac{kZe^2}{r^2}$ with the centripetal demand and substituting Bohr's quantisation $mvr = n\hbar$ gives $r_n = \tfrac{n^2 \hbar^2}{m k Z e^2}$, i.e. $r_n \propto \tfrac{n^2}{Z}$. Energy scales the opposite way: $E_n \propto -\tfrac{Z^2}{n^2}$.
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