Lenses and the eye

Your eye focuses on a book 25 cm away by raising its power from about 50 to 54 diopters, an 8% change.

Converging lenses and the focal point

A magnifying glass held in sunlight makes a tiny bright spot. The lens is thicker in the middle than at the edges, and its curved faces refract each ray by just the right amount so that rays arriving parallel to the axis all cross at one point. OpenStax calls that point the focal point F and its distance from the centre of the lens the focal length f. The word lens comes from the Latin for lentil, after the shape. A lens thicker in the middle is converging; one thicker at the edges is diverging and spreads parallel rays out as if they came from a focal point in front of it.

Opticians describe lenses by their power instead of the focal length: P = 1/f, measured in diopters (D) when f is in metres. A lens with a 0.25 m focal length has a power of 4 D. Stronger lenses have shorter focal lengths and larger powers, and spectacle prescriptions are written in diopters, positive for converging lenses and negative for diverging ones.

Finding an image with three rays

To find where a lens puts the image of an object, draw rays from the tip of the object using three rules from OpenStax. A ray parallel to the axis passes through the focal point on the far side. A ray through the centre of the lens does not change direction. A ray through the focal point on the near side comes out parallel to the axis. Where the rays cross is the tip of the image. With the object beyond the focal length the rays really meet behind the lens, and the image is real and upside down: it can be caught on a screen, as in a camera or a projector.

The thin lens equation

Ray diagrams give a picture; the thin lens equation gives numbers. With object distance dₒ, image distance dᵢ and focal length f, OpenStax writes 1/dₒ + 1/dᵢ = 1/f, and the magnification is m = −dᵢ/dₒ. In its worked example a bulb filament 0.750 m from a lens with f = 0.500 m gives 1/dᵢ = 1/0.500 − 1/0.750, so dᵢ = 1.50 m and m = −2.00: a real image twice the size and upside down (negative m). Bring the object inside the focal length and dᵢ comes out negative. That means a virtual image on the same side as the object, upright and larger. OpenStax’s magnifier example puts an object 7.50 cm from a 10.0 cm lens and finds dᵢ = −30.0 cm and m = 4.00.

The eye as a lens

OpenStax treats the eye as a single thin lens made of the cornea and the lens together, projecting a real, upside-down image onto the retina, about 2.00 cm behind. The biggest change in refractive index, and so most of the bending, happens where light enters the cornea from air (the cornea’s index is 1.38), not at the lens inside. The distance to the retina cannot change, so to focus at different distances the eye changes its power instead: muscles make the lens thicker and more strongly curved for near objects. For a 2.00 cm eye, P = 1/dₒ + 1/dᵢ gives 50.0 D for distant objects and 54.0 D at the usual near point of 25 cm. The brain turns the upside-down image the right way up.