Accommodation of the eye

Problem

The lens of an eye has a fixed retina-to-lens distance $q = 2\;\mathrm{cm}$. Calculate the focal length needed to focus on (a) a distant object at $p = 10\;\mathrm{m}$, (b) a book at $p = 30\;\mathrm{cm}$. Explain why reading for a long time tires the eyes.

Solution

Setup. The image always forms on the retina, so $q = 2\;\mathrm{cm} = 0{,}02\;\mathrm{m}$ is fixed. From the thin-lens equation:

$\frac{1}{f} = \frac{1}{p} + \frac{1}{q}$

(a) Distant object $p = 10\;\mathrm{m}$:

$\frac{1}{f} = \frac{1}{10} + \frac{1}{0{,}02} = 0{,}1 + 50 = 50{,}1\;\mathrm{m^{-1}}$

$\ev{f_a \approx 2{,}0\;\mathrm{cm}}$

(b) Nearby book $p = 0{,}30\;\mathrm{m}$:

$\frac{1}{f} = \frac{1}{0{,}30} + \frac{1}{0{,}02} = 3{,}33 + 50 = 53{,}33\;\mathrm{m^{-1}}$

$\ev{f_b \approx 1{,}87\;\mathrm{cm}}$

Why it tires the eyes. For the distant object a longer focal length is needed (a more “relaxed” lens); for the nearby book a shorter focal length is needed, i.e. a more curved lens. Reducing the focal length requires continuous contraction of the ciliary muscle: reading for a long time keeps this muscle under tension, hence the fatigue (asthenopia).

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Topics: Ottica Concepts: Lenti e specchi Skills: Analisi di casi limite e fantafisica