When the image lands far
A person holds a book far away to read it. Far signs look clear.
- This lens is too weak: near rays cross behind the retina.
- A convex lens in front moves the crossing onto the retina.
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Slide the near point. Add the lens. Read its power.
The rule
Hypermetropia is far-sightedness. Far objects look clear. Near objects blur. The near point is the closest distance the eye sees sharply. In hypermetropia it is farther than 25 cm.
Hypermetropia has two causes. The eye lens may be too weak. Then it bends the rays too little. Or the eyeball may be too short.
In hypermetropia, near objects focus behind the retina. A convex lens fixes it. The lens puts a 25 cm object's image at the near point.
A convex lens adds bending. So the rays cross sooner, on the retina.
Worked example: near point 40 cm. The lens formula is 1/f = 1/v − 1/u. The letters are distances from the lens:
- u: the object distance.
- v: the image distance.
- f: the focal length.
A distance on the object's side is negative.
The lens makes an object at 25 cm look 40 cm away. So u = −25 cm and v = −40 cm. Then 1/f = 1/v − 1/u = −1/40 − (−1/25) = −1/40 + 1/25. Use 200 as the common bottom number. That gives −5/200 + 8/200 = 3/200. So f = 200/3 cm = 2/3 m. Power = 1 ÷ (2/3) = +1.5 D.
The plus sign means a convex lens.
Shortcut: 1 ÷ 0.25 m = 4 D. So power = 4 − 100 ÷ n D. Here n is the near point in cm. For n = 40: 4 − 2.5 = +1.5 D.