Physics · Optics · Grade 7

The eye and spectacles

Why do “−3” spectacles suit one person but not another?

The situation

Aziz, a Grade 8 student, began to see the writing on the board blurred, but reads a book easily. His grandmother, on the contrary, holds the newspaper at arm's length to read it. The doctor prescribed glasses for both: “−3.75” for one and “+2.5” for the other. What do these numbers mean, and why doesn't one pair of glasses suit everyone?

Topic

Lenses, the eye and spectacles. Short sight and long sight: choosing spectacles

Key ideas

Thin lens formula
Optical power
Accommodation

Formulas

\frac{1}{f} = \frac{1}{d} + \frac{1}{d^{'}}

Thin lens formula: $d$ is the distance to the object, $d^{'}$ the distance to the image ( $d^{'} < 0$ — virtual image).

D = \frac{1}{f}

Optical power, in dioptres (1 D = 1 m⁻¹). For a converging lens $D > 0$ , for a diverging lens $D < 0$ .

D = D_{1} + D_{2}

The optical powers of lenses placed close together add: eye + glasses.

Γ = \frac{d^{'}}{d}

Linear magnification: a real image is inverted.

D_{s} = - \frac{1}{d_{f a r}}

Glasses for a short-sighted eye: they shift the image of a distant object to the eye's “far point”.

Theory

The eye — a living camera

Light is refracted mainly by the cornea (≈ 43 D) and the lens; the optical power of the whole eye is ≈ 60 D. A real, inverted, reduced image forms on the retina — the brain “turns it the right way up”. When looking at a near object the ciliary muscle makes the lens more convex — accommodation: the optical power increases by several dioptres.

Short sight and long sight

In myopia the eyeball is longer: the image of a distant object forms in front of the retina, and on the retina there is a blurred circle. A diverging lens ( $D < 0$ ) corrects it. In hypermetropia the eye is shorter and the image is behind the retina; a young eye overcomes this with accommodation, but looking at near objects becomes difficult — a converging lens ( $D > 0$ ) is needed. In the model, 1 mm of length is equivalent to an error of ≈ 2.6–2.8 D.

Age and the near point

The nearest point the eye can see clearly is the near point: $d_{n e a r} = 1 / (A_{m a x} + \dots)$ . With age the lens hardens and $A_{m a x}$ decreases (in the model on average $18.5 - 0.3 \cdot age$ , D). At 14 the near point is ≈ 7 cm, at 60 ≈ 2 m — this is presbyopia, and “+” glasses are needed for reading.

The blur circle

If the image does not fall on the retina, each point becomes a circle of diameter $c$ . Its angular size ≈ pupil diameter × defocus: with a 4 mm pupil an error of 1 D is ≈ 14 arcminutes, while the details of letters are ~1 arcminute. If the pupil shrinks, the circle shrinks too — which is why you see better when you squint. Simplifications: the eye is a single refracting surface ( $n^{'} = 1.336$ ), the power of the glasses is added directly, and only geometric blur is calculated.

The lens laboratory

Focal length of a converging lens: form a sharp image on the screen and measure $d$ and $d^{'}$ : $f = d d^{'} / (d + d^{'})$ , optical power $D = 1 / f$ ; take the mean over several $d$ (“Unknown A”). A diverging lens gives no real image by itself, so the image $S^{'}$ formed by an auxiliary converging lens is used as a virtual object for it: if lens B is placed a distance $a$ in front of $S^{'}$ , a new real image $S^{''}$ forms at a distance $b$ from it and $1 / f_{B} = 1 / b - 1 / a$ (the result is negative). The focal length of a lens depends on its refractive index and the radii of curvature of its surfaces: $1 / f = (n - 1) (1 / R_{1} + 1 / R_{2})$ (the radius of a convex surface is taken as positive, that of a concave surface as negative). The radius is measured by a spherometer: its three feet stand on a circle of radius $r$ , and the sagitta $h$ is read when the central screw touches the surface; from the right-angled triangle ${(R - h)}^{2} + r^{2} = R^{2}$ , i.e. $R = r^{2} / (2 h) + h / 2$ . The radius can also be found with Newton's rings (“Interference and diffraction”).

Mirrors and aberrations

A spherical mirror forms an image just as a lens does: $1 / f = 1 / d + 1 / d^{'}$ , $f = R / 2$ — car mirrors and headlights are in the “Car mirrors and the periscope” experiment. The image from a real lens is not ideal — there are aberrations: marginal rays are refracted more strongly than central ones (spherical), colours are focused at different places (chromatic), an oblique beam turns a point into a comma-shaped patch (coma), the rays of an oblique beam in two mutually perpendicular planes focus at different distances (astigmatism), the magnification changes towards the edge (distortion). The astigmatism of the eye is corrected with cylindrical glasses. Measuring and reducing aberrations — in the “Optical instruments” experiment; the lenses here are thin and paraxial, and aberrations are not included.

Did you know?

About two-thirds of the eye's optical power is provided by the cornea (≈ 43 D), the rest by the lens; the power of the whole eye is about 60 dioptres.
In youth the lens can increase its optical power by more than 10 dioptres; by the age of 50 this ability falls to a few dioptres — which is why most people need reading glasses after 40–45.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Car mirrors and the periscope — Grade 7
Laser, water and optical fibre — Grade 7
Shadow and eclipses — Grade 7
Measuring the speed of light — Grade 9
Optical instruments — Grade 9
Interference and diffraction — Grade 11
Polarisation of light — Grade 11
Laser physics and holography — Grade 11
Luminous flux and illuminance — Grade 11
Infrared, UV and X-rays — Grade 11