Question
A compound microscope has an objective lens of numerical aperture (NA) = 1.25, and uses light of wavelength λ = 550 nm. Using the Rayleigh criterion, find the minimum distance between two point objects that can just be resolved by this microscope. Also explain why increasing the NA improves resolution.
Solution — Step by Step
State the Rayleigh Criterion for a Microscope
The minimum resolvable distance (limit of resolution) for a microscope is:
Here, NA = n sin θ, where n is the refractive index of the medium between the object and objective lens, and θ is the half-angle of the cone of light collected. The 0.61 comes from the first zero of the Airy diffraction pattern — this is not arbitrary, it's the exact mathematical condition where two Airy discs are "just separated."
Plug in the Given Values
We have λ = 550 nm = 550 × 10⁻⁹ m and NA = 1.25:
So d ≈ 268 nm ≈ 0.27 μm.
Verify the NA Value Makes Physical Sense
NA = n sin θ. In air, n = 1, so NA ≤ 1. But here NA = 1.25 — this tells us the microscope uses oil immersion (typically n ≈ 1.515 for immersion oil). This is standard in high-power microscopy. The question is well-posed.
Answer the 'Why' Part — Effect of Increasing NA
Larger NA means either a larger collection angle θ or a denser medium (higher n). Since d ∝ 1/NA, doubling the NA halves the minimum resolvable distance — meaning we can distinguish features twice as fine. This is WHY oil immersion lenses exist: they push NA beyond 1.0, which air-based optics can never achieve.
Why This Works
When light from a point source passes through a circular aperture (the objective lens), it doesn't form a perfect point image — it spreads into an Airy disc due to diffraction. Two nearby objects each produce their own Airy disc. If those discs overlap too much, the brain can't tell there are two separate points.
The Rayleigh criterion says: two points are just resolved when the central maximum of one Airy disc falls exactly on the first minimum of the other. Mathematically, this gives the 0.61λ/NA formula. Smaller λ or larger NA both shrink the Airy disc, pushing the resolution limit down.
The numerical aperture captures both the geometry (sin θ) and the medium (n). Using oil immersion increases n without changing the physical setup, effectively making the wavelength of light inside the medium shorter — λ_medium = λ_air / n — which directly improves resolution.
Alternative Method — Using the Full Rayleigh Formula
Some textbooks write the resolving power as the reciprocal: RP = NA / (0.61λ). Resolving power is not a distance — it's the ability to resolve, so higher is better. This often confuses students because "resolving power" and "limit of resolution" are inverses of each other.
This form is useful when JEE asks "compare resolving powers of two microscopes" — just compare their NA/λ ratios directly. No need to compute actual distances.
💡 Expert Tip
JEE often gives two microscopes and asks which has better resolving power. Just compute NA/λ for each. The one with higher NA/λ wins. If wavelengths are equal, higher NA always wins.
Common Mistake
⚠️ Common Mistake
Students confuse resolving power (a number, higher = better) with limit of resolution (a distance, smaller = better). In JEE Advanced 2023, several students lost marks by writing that "higher resolving power means the microscope can see objects farther apart" — exact opposite of correct. Resolving power = 1 / (limit of resolution). Keep this straight before the exam.
A second common slip: using the telescope formula (1.22λ/D) instead of the microscope formula (0.61λ/NA). The 1.22 vs 0.61 difference is real — for a circular aperture (telescope), the geometry gives 1.22; the microscope formula accounts for the cone of light from both sides, effectively halving the constant. Don't mix them up in the same paper — NEET and JEE both ask both types.