The magnifying glass

The simplest of optical combinations is the magnifying glass: a converging lens placed at a distance $p < f$ from the object. Under these conditions the refracted rays still diverge after the lens, and the eye sees a virtual image, upright and magnified, on the same side of the lens as the object.

What really matters, when looking at something, is not the absolute size of the image but the angle under which it is seen: the larger the angle, the more detail the eye can make out. For this reason the angular magnification is defined as the ratio between the angle $\theta'$ under which the image appears through the lens and the angle $\theta$ under which the object appears to the naked eye, placed at the minimum accommodation distance $d_0 = 25$ cm (the standard reading distance). With a bit of geometry one finds:

Key formula

$\ev{G = \dfrac{\theta'}{\theta} \approx 1 + \dfrac{d_0}{f}}$

The formula holds for lenses held close to the eye; if the eye is “relaxed” and focused at infinity, one instead has $G = d_0/f$. In both cases the magnification grows as the focal length decreases.

Jeweller's loupes

This is why jewellers’ or fine-work loupes have focal lengths $f \sim 5$–$15$ mm: the magnification $G$ ranges from about $18\times$ to about $5\times$.

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Topics: Ottica Concepts: Lenti e specchi

Related exercises: Problema — Raggi per oggetto oltre 2f (lente) · Problema — Raggi per specchio concavo con p minore di f · Esercizio svolto — Specchio concavo