Resolving Power And Diffraction Limit

The Rayleigh criterion sets the diffraction limit on resolution for a circular aperture: two point sources are just resolved when the central maximum of one falls on the first diffraction minimum of the other, giving minimum angular separation ${\theta }_{min}=\frac{1.22\lambda }{D}$, where $\lambda$ is the wavelength and $D$ the aperture diameter. Substituting $\lambda =500\times 10^{-9}$ m and $D=0.1$ m yields ${\theta }_{min}=\frac{1.22\times 500\times 10^{-9}}{0.1}=\frac{6.1\times 10^{-7}}{0.1}=6.1\times 10^{-6}$ rad. The factor 1.22 is not arbitrary—it emerges from the first zero of the Bessel function describing diffraction by a circular aperture, replacing the simple $\lambda /a$ that applies to a rectangular slit. Resolution improves (the resolvable angle shrinks) either by using shorter wavelengths or by enlarging the aperture, which is the central reason astronomers build ever-larger primary mirrors and why ultraviolet or X-ray instruments can resolve finer structure than visible-light ones. The value $5.0\times 10^{-6}$ rad is wrong because it omits the 1.22 circular-aperture factor. The value $1.22\times 10^{-6}$ rad is wrong as it drops the wavelength scaling. The value $6.1\times 10^{-5}$ rad is wrong due to a misplaced power of ten in the aperture conversion. This NCERT limit explains why larger telescopes resolve finer detail. A check confirms a bigger aperture lowers ${\theta }_{min}$, so the micro-radian result is sensible for a 10 cm lens.