Q.The human eye has an approximate angular resolution of $\phi = 5.8 \times 10^{-4}\ \text{rad}$ and a typical photoprinter prints a minimum of 300 dpi (dots per inch, $1\ \text{inch} = 2.54\ \text{cm}$). At what minimal distance $z$ should a printed page be held so that one does not see the individual dots.

The Core Problem: When Two Points Become One Blur

Imagine you are looking at two tiny dots drawn very close together on a piece of paper. From far away, they look like a single dot. As you bring the paper closer, at some point your eye suddenly sees two separate dots. That moment — the threshold where your eye (or a microscope) can just barely tell that there are two objects instead of one — is the heart of resolving power.

A microscope's job is to show you fine detail. But no matter how good the lenses are, there is a fundamental limit: light itself behaves like a wave. When light passes through the circular opening of a lens, it does not travel in perfect straight lines. It spreads out and forms a pattern called an Airy disk — a bright central spot surrounded by faint rings. Every point in your specimen becomes a tiny blurry disk in the image, not a perfect point.

If two points in the specimen are very close, their Airy disks overlap. When they overlap too much, your eye cannot tell them apart — they merge into one blob. The resolving power of a microscope is its ability to show two closely spaced points as distinct.

The Precise Criterion: Lord Rayleigh's Condition

Lord Rayleigh proposed a practical rule: two points are just resolved when the centre of one Airy disk falls exactly on the first dark ring of the other. At that point, the combined intensity has a small dip between the two peaks — your eye can just detect that there are two sources.

For a microscope, the smallest distance $d$ between two points that can just be resolved is given by:

where:

• $\lambda$ is the wavelength of light used
• $n$ is the refractive index of the medium between the specimen and the objective lens
• $\beta$ is the half-angle of the cone of light entering the objective

The quantity $n \sin \beta$ is called the numerical aperture (NA) of the objective lens. So the formula is often written as:

What This Tells Us: Two Levers for Better Resolution

1. Shorter wavelength $\lambda$ — Blue light resolves better than red light. Ultraviolet light resolves even better, which is why electron microscopes (using much shorter "wavelengths" of electrons) can see atoms.

2. Larger numerical aperture $n \sin \beta$ — You can increase $n$ by using oil between the slide and the objective (oil immersion). Air has $n \approx 1$, but special oils have $n \approx 1.5$. You can increase $\sin \beta$ by using a lens that collects light from a wider cone — a lens with a shorter focal length and larger diameter. …