 Solution

For a diffraction grating with $N$ illuminated lines in order $p$, its ideal spectral resolving power isThe 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 requiresWith line density $500\mathrm{m}{\mathrm{m}}^{-1}$, the ideal continuous estimate for the illuminated width isRounding the required number of illuminated grating lines upward gives $117$ lines, conventionally an illuminated width $W\simeq 117/(500\mathrm{m}{\mathrm{m}}^{-1})=\boxed{0.234\mathrm{mm}}$. 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.