= Solution
The image annotation gives probe <electric current> $I=43\,\mathrm{pA}=4.3\times10^{-11}\,\mathrm A$ and working distance $L=2.4\,\mathrm{mm}$. Interpret $d_o$ as the probe diameter and $\alpha$ as its convergence semiangle, in radians. To estimate that angle from the available dimensions, approximate the effective aperture plane as a distance $L$ from the specimen. With aperture radius $r_a=25\,\mathrm{\mu m}$,
$$
\alpha\simeq\frac{r_a}{L}=1.0417\times10^{-2},\qquad d_o=\sqrt{\frac{4I}{\pi^2\alpha^2\beta}}=\boxed{4.01\,\mathrm{nm}}.
$$
The <electron beam brightness> $\beta$ measures <electric current> per probe area per <solid angle>; the supplied convention corresponds to a circular probe and a paraxial cone of <solid angle> approximately $\pi\alpha^2$. The aperture diameter must be halved, and the annotated current is in picoamperes. Treating the full aperture angle as $\alpha$ would introduce a factor-of-two error in $d_o$.
This is a conditional optical estimate: the working distance is not necessarily the actual electron-optical distance from the effective aperture to the focused probe. The answer assumes that approximation, the quoted <electron beam brightness> at the operating <voltage>, and negligible <optical aberrations>. The 30 kV annotation gives a much shorter <de Broglie wavelength> than 4 nm, so a nanometre-scale probe is plausible. Actual <electron microscope resolution> also depends on <diffraction>, scattering in the specimen, signal generation and collection, <signal-to-noise ratio>, drift and specimen charging. A 4 nm probe does not establish that every measured edge width is 4 nm.
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