= Solution
For a <diffraction grating> with $N$ illuminated lines in order $p$, its ideal <spectral resolving power> is
$$
\mathcal R=\frac\lambda{\Delta\lambda}=pN.
$$
The two <spectral lines> have separation $\Delta\lambda=3.4\,\mathrm{nm}$ and mean <wavelength> $\overline\lambda=395.1\,\mathrm{nm}$. In first order, $p=1$, so the <Rayleigh criterion> requires
$$
N\gtrsim\frac{395.1}{3.4}=116.2.
$$
With line density $500\,\mathrm{mm^{-1}}$, the ideal continuous estimate for the illuminated width is
$$
W\simeq\frac{116.2}{500\,\mathrm{mm^{-1}}}=\boxed{0.2324\,\mathrm{mm}\simeq0.23\,\mathrm{mm}.}
$$
Rounding the required number of illuminated grating lines upward gives \b[$117$ lines], conventionally an illuminated width $W\simeq117/(500\,\mathrm{mm^{-1}})=\boxed{0.234\,\mathrm{mm}}$. Using the longer <wavelength> rather than the mean also gives the same integer requirement. This is the ideal limit set by the <resolving power of a diffraction grating>; additional broadening from the illumination or observing apparatus could require a larger width.
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