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 human eye has an approximate angular resolution of φ = 5.8 × 10⁻⁴rad and a typical photoprinter prints a m

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\phi = 5.8 \times 10^{-4}\ \text{rad}

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Concept understanding — Resolving Power of Microscope

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.

Note

Resolving power is not about magnification. You can magnify a blurry image as much as you like — it only becomes a bigger blur. Resolution is about separating detail, not enlarging it.

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:

d = \frac{1.22 \lambda}{2 n \sin \beta}

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:

d = \frac{1.22 \lambda}{2 \cdot \text{NA}}

\text{Resolving power} = \frac{1}{d} = \frac{2 \cdot \text{NA}}{1.22 \lambda}

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. …

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The key idea is to match the printer’s dot spacing to the eye’s angular resolution limit. The minimal viewing distance is $z \approx 14.6\ \text{cm}$ .

Why this approach works

When you look at a printed page, your eye can only resolve two points if the angle between them is larger than about $5.8 \times 10^{-4}$ radians. If the dots are closer together than that angular limit, they blur into a continuous tone — that’s exactly how halftone printing tricks the eye into seeing smooth shades.

The printer puts down 300 dots per inch. That means the centre-to-centre distance between adjacent dots is fixed by the printer’s resolution. The question asks: how far away must you hold the page so that the angle subtended by that dot spacing just equals the eye’s resolution limit? Any closer and you’d see the individual dots; any farther and they merge.

Step-by-step solution

1. Find the dot spacing from the printer’s DPI

300 dpi means 300 dots in every inch. So the distance between two neighbouring dot centres is:

d = \frac{1\ \text{inch}}{300} = \frac{2.54\ \text{cm}}{300}

d = 8.467 \times 10^{-3}\ \text{cm}

That’s about $0.085\ \text{mm}$ — very fine, but still resolvable if you bring the page close enough.

2. Relate dot spacing to angular resolution

For small angles (which this is), the angle $\phi$ subtended by an object of size $d$ at distance $z$ is:

\phi \approx \frac{d}{z}

This approximation is excellent when $\phi$ is in radians and $d \ll z$ , which holds here.

3. Set the angle equal to the eye’s resolution limit

We want the dots to be just unresolvable, so:

\frac{d}{z} = \phi

Substitute the numbers: …

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Concept understanding — Resolving Power of Microscope

The Core Problem: When Two Points Become One Blur

The Precise Criterion: Lord Rayleigh's Condition

What This Tells Us: Two Levers for Better Resolution

Why this approach works

Step-by-step solution

Method: Finding the Minimum Distance for Two Points to Be Just Resolved by Eye

Steps

Why this approach works

Step-by-step solution

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