Question bank — Resolving power — Rayleigh criterion

True — ${\theta }_R=1.22\lambda /D$, so increasing $D$ shrinks the resolvable angle. A wider aperture diffracts less, so each Airy disc is tighter and two sources stay distinguishable at smaller separations.

False — they are two separate gains. Light-gathering scales with area ($\propto D^2$, brightness), while resolution scales with $1/D$ (fineness of detail); a big telescope wins on both but for different physical reasons.

False — it is an agreed convention marking where the combined intensity dip (~26%) becomes detectable. Nature does not "switch" from one blob to two at exactly $1.22\lambda /D$; the criterion just draws a fair, computable line.

False — the 1.22 is only for a circular aperture (first zero of the Bessel function $J_1$). A slit uses the plain ${\theta }_R=\lambda /a$; the factor comes from integrating over a disc, not a strip.

True — $d_{\min }\propto \lambda$, so shorter-wavelength (blue) light gives a smaller resolvable distance. This is exactly why UV and electron microscopes (tiny effective $\lambda$) resolve far finer detail.

False — beyond the diffraction limit you only enlarge a blur ("empty magnification"). Detail is capped by ${\theta }_R$ or $d_{\min }$, set by diffraction, not by how much the image is stretched.

False (missing the point) — transparency is necessary but the real gain is that oil's higher refractive index $n$ raises the N.A. $=n\sin \beta$, widening the collecting cone and shrinking $d_{\min }=0.61\lambda /\text{N.A.}$

False — Rayleigh is a human-eye convention; a low-noise detector plus curve-fitting can separate peaks somewhat closer. The criterion marks "comfortably resolvable," not an absolute wall.

True — ${\theta }_R=1.22\lambda /D$ depends on the ratio $\lambda /D$. Doubling both gives $(2\lambda )/(2D)=\lambda /D$, so the effects cancel exactly.

The formula returns radians, not degrees. Only in radians does the small-angle relation $s=\theta L$ give the correct physical spacing on the retina or page.

Wrong direction — a smaller $D$ means more diffraction spreading (${\theta }_R\propto 1/D$), so a constricted pupil resolves slightly worse. The benefit of a small pupil is depth of field, not resolution.

The circular geometry pushes the first zero outward by the factor 1.22, giving ${\theta }_R=1.22\lambda /D$. Dropping the 1.22 uses the slit result for a disc — a common slip.

Moving the sample away shrinks the cone half-angle $\beta$, lowering N.A. $=n\sin \beta$ and increasing $d_{\min }$. You want the sample close so the objective subtends a wide cone.

Reversed — green has a shorter $\lambda$, so ${\theta }_R$ is smaller and the same spacing is now comfortably resolvable. Shorter wavelength always helps.

Backwards — resolving power is $1/{\theta }_R$. A small ${\theta }_R$ (fine angle) means high resolving power; a large ${\theta }_R$ means the instrument can only separate coarse, widely-spaced sources.

A finite hole chops the edges off the wavefront, and by Huygens' principle every truncated wavefront spreads — chopping a wave is diffraction, so no finite lens can focus a point to a true point.

A slit integrates diffraction over a 1-D strip (first null at $\lambda /a$); a circle integrates over a 2-D disc, giving a Bessel-function pattern whose first zero $J_1$ sits at $1.22\lambda /D$ — the 2-D geometry redistributes energy outward.

Because the first-minimum position is already known exactly, making the criterion clean to compute, and at that spacing the ~26% central dip roughly matches what a typical eye can just detect.

With $\lambda \approx 500$ nm and the best oil N.A. $\approx 1.4$, $d_{\min }=0.61\lambda /\text{N.A.}\approx 220$ nm. You cannot beat this with ordinary lenses because it is set by diffraction, not lens quality.

Not only for light-gathering — larger $D$ directly shrinks ${\theta }_R=1.22\lambda /D$, letting them separate stars (and planet-star pairs) that a small aperture merges into one blob.

The arc-length relation $s=\theta L$ is only valid in radians (it comes from $s=r\theta$ for a circle). Degrees would introduce a spurious factor of $\pi /180$.

${\theta }_R=1.22\lambda /D\to 0$ — an infinitely wide aperture would focus a point to a true point with zero diffraction blur, the ideal (unreachable) limit.

${\theta }_R\to 0$, so arbitrarily fine detail becomes resolvable — this is why electron microscopes, with picometre-scale de Broglie wavelengths, vastly outperform light microscopes.

Two peaks separated by a shallow central dip of about 26% below the peaks — just deep enough to signal "two," so it sits precisely on the borderline between resolved and merged.

$d_{\min }=0.61\lambda /1\approx 305$ nm — worse than oil immersion's ~220 nm, showing that raising $n$ above 1 is the only way to push finer without changing $\lambda$.

No — they remain two distinct sources; only their diffraction discs overlap so heavily that the central dip vanishes and the detector records a single peak. Resolution is a limit of the instrument, not of the objects.