Resolving Power in Optics
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Direct answer
Two point sources are just resolved when the central maximum of one's diffraction pattern falls on the first minimum of the other — Rayleigh's criterion. For a telescope objective of diameter D, the minimum resolvable angle is θ_min = 1.22λ/D, so resolving power (defined as 1/θ_min) grows with aperture and shrinks with wavelength. For a microscope, the minimum separable distance is d = 0.61λ/(μ sin θ), so resolving power grows with numerical aperture μ sin θ and with shorter wavelengths — the logic behind oil-immersion objectives and ultraviolet or electron microscopes. The human eye resolves about one arc-minute (2.9 × 10⁻⁴ rad), the benchmark every "can the eye separate this" question runs against.
What you must remember
- Rayleigh criterion: θ_min = 1.22λ/D for a circular aperture — the 1.22 is the first zero of the Airy pattern; slit apertures use sin θ = λ/a instead.
- Telescope resolving power: RP = D/(1.22λ); doubling the objective diameter halves the smallest resolvable angle — the reason observatories chase large mirrors, not magnification.
- Microscope resolving power: RP = 2μ sin θ/λ = 2 NA/λ with numerical aperture NA = μ sin θ; the smallest resolvable separation is d = 0.61λ/NA, about half the wavelength at best with oil.
- Oil immersion: cedar oil (μ ≈ 1.5) between objective and slide raises NA from about 1 (dry air) to about 1.5, cutting d by a third — the standard high-resolution configuration of biology-style microscopes.
- Eye benchmark: one arc-minute, roughly 2.9 × 10⁻⁴ rad, set by the ~2-3 mm pupil at 550 nm; two headlights 1.2 m apart merge beyond about 4 km (1.2/2.9 × 10⁻⁴).
- Wavelength lever: RP ∝ 1/λ, so electron microscopes using picometre de Broglie wavelengths resolve atoms that visible light (hundreds of nanometres) never can.
- Definition discipline: resolving power is the reciprocal of the minimum resolvable quantity (angle for telescopes, distance for microscopes) — options deliberately quote d where RP is asked.
Headlights, stars and oil
How far can the eye separate two car headlights 1.2 m apart? Set the separation equal to the eye's resolution angle times distance: 1.2 = 2.9 × 10⁻⁴ × d, giving d ≈ 4.1 km — beyond that, on a perfectly dark straight road, the pair merges into one glow. Run the same logic upward in scale: a double star separated by 10⁻⁶ rad needs D = 1.22λ/θ = 1.22 × 550 × 10⁻⁹/10⁻⁶ ≈ 0.67 m, so a 67 cm objective just resolves it in green light — and the traditional Indian undergraduate observatories with 38 cm telescopes stop short, while modern 2 m-class instruments breeze past. Magnification enlarges the image but manufactures no new information; the objective's diameter alone decides what was ever separable.
The microscope side trades aperture for medium. With green light (550 nm) and a dry objective NA = 1.0, d = 0.61 × 550/1.0 ≈ 336 nm — a cell organelle of 300 nm is at the edge. Fill the object space with immersion oil of μ = 1.5 and sin θ near 0.9: NA = 1.35, and d falls to about 250 nm. To go further, shorten the wavelength itself: ultraviolet microscopy and, decisively, the electron microscope, whose 0.1 nm-class de Broglie waves turned viruses and macromolecules from "invisible" into textbook figures. The ladder of resolution — aperture, medium, wavelength — is exactly the order in which JEE wants the reasoning written.
Where students slip
The definitional trap heads the list: resolving power is 1/d or 1/θ_min, so a question asking which instrument has greater resolving power has answers opposite to one asking which resolves the smaller separation. Second, the 1.22 factor belongs to circular apertures; a single-slit question using it (or vice versa) loses a mark quietly. Third, telescope questions tempt with eyepiece data — magnification and eyepiece focal length are irrelevant to θ_min, which the objective sets alone. In microscope problems, remember μ sits inside NA (oil immersion raises it), and the formula's λ is the wavelength in vacuum (or in the illumination), not in the oil. A subtle Advanced-level favourite: RP of a diffraction grating (nN, the order times total lines ruled) is a different quantity from image resolution, and matching-the-following columns have juxtaposed the two to catch formula grab-bags. Finally, the eye's one arc-minute is worth memorising with its radian value — questions convert it silently.
Frequently asked questions
What is Rayleigh's criterion for resolution?
Two images are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other, giving θ_min = 1.22λ/D for a circular aperture.
How can a telescope's resolving power be increased?
By enlarging the objective diameter D, since RP = D/(1.22λ); eyepiece changes alter magnification only, not the smallest resolvable angle.
Why does an oil-immersion objective resolve better?
The oil's refractive index raises the numerical aperture NA = μ sin θ, shrinking the minimum resolvable distance d = 0.61λ/NA — roughly a one-third improvement over a dry objective.
What is the resolving power of the human eye?
About one arc-minute, 2.9 × 10⁻⁴ rad, fixed by a 2-3 mm pupil at visible wavelengths — two points closer than this angle merge into one.
Why do electron microscopes resolve atoms while optical microscopes cannot?
Resolution scales with wavelength, and electrons' de Broglie wavelengths are picometres against hundreds of nanometres for light, improving the minimum resolvable distance by a factor of about a million.