= Solution
{c}
The printed <magnetic dipole moment> uses dimensionless <angular momentum operators>: their physical counterparts are $\hbar\mathbf L$ and $\hbar\mathbf S$. Choose the <magnetic field> along $z$. Then the <Zeeman energy> is $H_Z=\mu_BB(L_z+g_sS_z)$, whose characteristic size is $\mu_BB$. In <LS coupling>, treating this as a <time-independent perturbation theory> requires it to be much smaller than the relevant <fine structure> separation, rather than merely smaller than the <Coulomb interaction> <energy>.
For a concrete <hydrogen atom> estimate take the $2p_{1/2},2p_{3/2}$ pair. The <hydrogen fine-structure correction> is
$$
E_{nj}=-\frac{\mathrm{Ry}}{n^2}-\frac{\mathrm{Ry}\,\alpha_{\rm fs}^2}{n^3}\left(\frac1{j+1/2}-\frac3{4n}\right),
$$
where $\alpha_{\rm fs}$ is the <fine-structure constant>, distinguished from the later potential coefficient. Thus
$$
\Delta E_{2p}=\frac{\mathrm{Ry}\,\alpha_{\rm fs}^2}{16}\simeq4.53\times10^{-5}\,\mathrm{eV},\qquad
\frac{\Delta E_{2p}}{\mu_B}\simeq0.78\,\mathrm T.
$$
Consequently a suitable weak-<magnetic field> condition for this pair is
$$
\boxed{B\ll0.8\,\mathrm T;\quad B\sim10^{-2}\,\mathrm T\text{ is comfortably perturbative.}}
$$
There is no single threshold for every <hydrogen> <energy level>: the relevant <fine structure> separations decrease with $n$. The $1s$ <energy level> has no same-$n$ <spin-orbit interaction> partner, so it cannot itself define this comparison. This calculation neglects <hyperfine structure>; resolving that smaller scale requires an additional weak-<magnetic field> condition.
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