= Solution
{c}
Treat each grain as an isolated compact sphere and take its wavelength-independent <radiation-pressure efficiency> to be $Q_{\rm pr}=1$. At distance $R$, its mass and radiation-force cross-section are
$$
m_d=\frac43\pi r_d^3\rho_d,\qquad A_d=\pi r_d^2.
$$
The inward gravitational force and outward <radiation pressure> force are
$$
F_G=\frac{GM_{\rm BH}m_d}{R^2},\qquad
F_{\rm rad}=\frac{L}{4\pi R^2c}A_d.
$$
Thus the opacity per unit dust mass and the <dust-grain Eddington limit> are
$$
\kappa_d=\frac{A_d}{m_d}=\frac{3}{4\rho_dr_d},\qquad
L_{\rm Edd,d}=\frac{4\pi GM_{\rm BH}c}{\kappa_d}.
$$
To equal the <Eddington luminosity> based on <electron-scattering opacity>, $\kappa_d$ must equal $\kappa_{\rm es}$. Therefore
$$
\boxed{r_{d,\rm crit}=\frac{3}{4\rho_d\kappa_{\rm es}}.}
$$
For fully ionized hydrogen, $\kappa_{\rm es}=\sigma_T/m_p\simeq0.40\,\mathrm{cm^2\,g^{-1}}$, and the stated $\rho_d=1\,\mathrm{g\,cm^{-3}}$ gives
$$
\boxed{r_{d,\rm crit}\simeq1.9\,\mathrm{cm}.}
$$
Using $0.34\,\mathrm{cm^2\,g^{-1}}$ for a roughly solar-composition gas instead gives $2.2\,\mathrm{cm}$. This composition-dependent normalization should be stated because the problem does not fix it. For a different constant <radiation-pressure efficiency>, the critical radius is multiplied by $Q_{\rm pr}$.
At $L=L_{\rm Edd,es}$, the grain's <radiation-pressure coefficient> is
$$
\frac{F_{\rm rad}}{F_G}=\frac{\kappa_d}{\kappa_{\rm es}}
=\frac{r_{d,\rm crit}}{r_d}.
$$
Consequently, \b[smaller grains experience a net outward force, larger grains remain gravitationally attracted, and critical-size grains have zero net radial force]. Grains released at rest are therefore size-sorted into expelled and infalling populations. With an initial inward velocity, an outward force does not guarantee reversal before the grain reaches the hole; trajectories also depend on initial conditions. Gas drag and dust destruction have been omitted from this isolated-grain calculation. Ordinary interstellar grains are much smaller than a centimetre, so their direct radiative acceleration is very strong in this idealized model.
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