= Solution
The <Bohr model> combines classical circular motion in the <Coulomb potential> with explicitly quantum postulates. A one-electron atom of nuclear charge $+Ze$ has electron–nucleus <reduced mass> $\mu$. With $C=Ze^2/(4\pi\varepsilon_0)$, the relative circular orbit obeys
$$
\frac{\mu v^2}{r}=\frac C{r^2},\qquad L=\mu vr.
$$
Bohr postulates permitted nonradiating stationary orbits with $L=n\hbar$, for $n=1,2,\ldots$. These assumptions give
$$
\boxed{r_n=\frac{n^2\hbar^2}{\mu C}
=\frac{4\pi\varepsilon_0\hbar^2}{\mu Ze^2}n^2.}
$$
For hydrogen with the nucleus taken infinitely heavy, the $n=1$ radius is the <Bohr radius>, $a_0\simeq5.29\times10^{-11}\,\mathrm m$. The finite <reduced mass> gives the small nuclear-motion correction, and increasing $Z$ contracts the one-electron orbit.
The circular-motion equation gives $T=C/(2r)$ while the <potential energy> is $-C/r$, so the total <energy> is $-C/(2r)$. Substituting the quantized radius yields
$$
\boxed{E_n=-\frac{\mu C^2}{2\hbar^2n^2}
=-\frac{\mu Z^2e^4}{2(4\pi\varepsilon_0)^2\hbar^2n^2}.}
$$
For the heavy-nucleus hydrogen approximation, $E_n\simeq-13.6\,\mathrm{eV}/n^2$. The ionization threshold is zero <energy>, so removing the ground-state electron requires about $13.6\,\mathrm{eV}$.
The model further postulates that radiation is emitted or absorbed only in transitions between stationary orbits, with <photon energy> $h\nu=|E_i-E_f|$. Emission from $n_i>n_f$ gives
$$
\frac1\lambda=R_\mu Z^2\left(\frac1{n_f^2}-\frac1{n_i^2}\right),\qquad
R_\mu=\frac{\mu e^4}{8\varepsilon_0^2h^3c}.
$$
Thus it reproduces the principal <hydrogen spectral series>, their limiting <wavelengths>, and the gross hydrogenic <energy> levels. For example, $n=3\to2$ in hydrogen has $\Delta E\simeq1.89\,\mathrm{eV}$ and $\lambda\simeq656\,\mathrm{nm}$, a visible line. Absorption reverses the transition when the photon matches the <energy> gap.
The achievement is that quantized energies explain discrete spectra, but the model's stationary orbits are postulates rather than consequences of classical electrodynamics: an accelerating classical charge would radiate. It does not provide a general calculation of line intensities or transition selection rules and fails as a quantitative theory of many-electron atoms. <Spin>, <fine structure> and the full <Zeeman effect> require further physics. Modern <quantum mechanics> replaces definite electron orbits by <wavefunctions> and quantum probabilities. In particular, the true hydrogen <ground state> has $\ell=0$, so its <orbital angular momentum> vanishes; the model's rule $L=n\hbar$ should not be identified with the modern orbital <eigenvalue> rule $L^2=\ell(\ell+1)\hbar^2$. \b[Bohr's <energy> quantization remains a successful hydrogenic approximation, while its literal orbital picture does not.]
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