Resolution Limit in a Teaching Microscope
A teaching laboratory has a brightfield microscope with a 40× objective of numerical aperture (NA) 0.65 and a dry condenser of NA 0.55. Illumination is filtered to 550 nm. A slide contains fixed cocci that appear in pairs; the two cells in a pair are separated by a gap of 0.45 µm between centres. A student reports seeing two distinct dots in every pair and plans to publish a counts table. The laboratory also owns a 100× oil-immersion objective of NA 1.30 and immersion oil of refractive index 1.515. The 100× objective has a working distance of 0.13 mm, and the slide is 1.1 mm thick. The student asks whether switching objectives will change the count and whether the images can be used for publication.
1. Use the Abbe criterion to decide whether the 40×/0.65 objective can resolve the 0.45 µm separation. Does the student's report stand?
Read the worked answer
Abbe resolution is d = 0.61λ/NA. With λ = 550 nm and NA 0.65, d = 0.61 × 550 / 0.65 = 516 nm, or about 0.52 µm. The separation is 0.45 µm, which is smaller than the resolution limit, so the pair should not be reliably resolved as two distinct cells. The student's claim of seeing two dots in every pair is therefore suspect.
The condenser matters as well. A dry condenser of NA 0.55 limits the effective NA to about 0.55, giving d = 0.61 × 550 / 0.55 = 610 nm, even worse. The student may be seeing diffraction artefacts, unequal staining, or two dots where the pair is oriented favourably. Before accepting the counts, the images should be checked against a calibration slide and scored blind. The report as written does not stand.
2. Determine the minimum NA needed to resolve 0.45 µm at 550 nm, and decide whether the oil-immersion objective meets it. Explain the role of oil and the condenser.
Read the worked answer
Rearranging d = 0.61λ/NA gives NA = 0.61λ/d = 0.61 × 550 / 450 = 0.746. The 40×/0.65 objective is below this threshold. The 100×/1.30 oil-immersion objective gives d = 0.61 × 550 / 1.30 = 258 nm, comfortably below 0.45 µm, so it meets the requirement.
Oil works by replacing the air gap between the objective and the slide with a medium of refractive index close to glass. This prevents total internal reflection and allows rays at higher angles to enter the objective, raising NA and light collection. For full benefit, the condenser should also be oiled or have a high NA; otherwise the illumination cone, not the objective, limits resolution. The 0.13 mm working distance is compatible with a standard coverslip, but the student must avoid crushing the slide and must clean oil from the lens after use.
3. Advise the student on whether the published counts should come from the 40× or the 100× objective, and what validation would make the count defensible.
Read the worked answer
The counts should come from the 100×/1.30 oil-immersion objective, because only that objective resolves the 0.45 µm separation at 550 nm. Magnification alone does not create resolution: increasing from 40× to 100× without sufficient NA would produce empty magnification, enlarging blurred images rather than separating the cells.
Validation should include a reference slide with spheres of known spacing, a blinded second reader, and replicate counts from at least two independently prepared slides. The student should record the objective, NA, condenser setting and illumination wavelength, because these determine the resolution limit. If the count must distinguish pairs at 0.45 µm near the limit, a higher-NA objective or electron microscopy may be needed. The publication should state the resolution limit and avoid implying that every pair was individually resolved when the separation is close to the Abbe limit.