Two Pairs of Spectacles in a Tailoring Shop
In a tailoring shop in a small town, Ramu, aged 48, complains that he must hold a needle about 50 cm from his eye to thread it; at 25 cm the thread blurs. His apprentice Sita, aged 15, can read only the largest letters on the blackboard from her seat 2 m away, and says she sees the board clearly from about 1.5 m. An optician's shop nearby sells ready-made reading spectacles of +2.0 D and distance spectacles of −0.75 D. Ramu, who likes saving money, suggests he will simply borrow Sita's distance spectacles for close work, and Sita will use his reading spectacles while walking home. Neither is a good plan, and the optician asks you, as the science student in the shop, to explain the calculations and the risks before either pair is bought.
1. Calculate the power of the lens Ramu needs for reading at the normal near point of 25 cm, and decide whether the ready-made +2.0 D pair is appropriate.
Read the worked answer
Ramu can focus comfortably only on objects at 50 cm; his near point has receded from the normal 25 cm. A reading lens must take an object placed at 25 cm and form its virtual image at 50 cm, where his eye can still focus.
Using 1/f = 1/v − 1/u with the object 25 cm in front of the lens and the image 50 cm on the same side:
u = −25 cm, v = −50 cm
1/f = (−1/50) − (−1/25) = −1/50 + 2/50 = +1/50
so f = +50 cm = +0.50 m and P = 1/f (in metres) = 1/0.50 = +2.0 D.
The lens is convex (converging), which is exactly the +2.0 D reading spectacle available. It should be used as intended - for near work only - because a converging lens blurs distant vision: parallel rays from a distant object would be brought to a focus before reaching his retina.
The calculation assumes the lens sits close to the eye and that he reads at 25 cm. If he prefers to read at 30–35 cm, a slightly weaker lens (+1.5 D) would be more comfortable, which is why an optician measures the working distance rather than guessing.
2. Sita's far point is 1.5 m. Calculate the power she needs and explain what her eye is doing to distant objects without a lens.
Read the worked answer
Sita cannot focus on anything beyond 1.5 m, so rays from infinity must be made to diverge as if they came from a point 1.5 m in front of her eye. For a distant object, u = −∞ and v = −1.5 m:
1/f = 1/v − 1/u = (−1/1.5) − 0 = −0.667
so f = −1.5 m and P = −0.67 D. Ready-made lenses come in 0.25 D steps, so −0.75 D is the closest available; it slightly over-corrects but is tolerable at this small error.
What her eye is doing: in myopia the eyeball is too long or the eye lens too strongly converging, so parallel rays from a distant object are brought to a focus in front of the retina. By the time the rays reach the retina they have crossed and the image is blurred. A concave (diverging) lens spreads the rays slightly before they enter the eye, effectively pushing the focus back onto the retina.
This is why she sees clearly at 1.5 m without help: a nearby object's rays are already diverging strongly, and the eye's excess converging power brings them to a focus exactly on the retina.
3. Ramu proposes to use Sita's −0.75 D spectacles for threading a needle at 25–30 cm. Show numerically what happens to his near point, and explain why Sita must not wear his +2.0 D pair for walking.
Read the worked answer
Ramu's eye can focus objects as close as 50 cm. With Sita's −0.75 D lens in front of it, the lens must form a virtual image at 50 cm from the eye for any object he intends to see. Using 1/f = 1/v − 1/u with f = 1/(−0.75) = −1.333 m and v = −0.50 m:
−0.75 = 1/(−0.50) − 1/u = −2 − 1/u
1/u = −2 + 0.75 = −1.25, so u = −0.80 m.
His effective near point is pushed out from 50 cm to 80 cm. The diverging lens makes every object seem farther away than it is, so his eye must converge even harder - exactly the opposite of what a presbyopic eye needs. Threading a needle at 25 cm would become impossible rather than easier.
For Sita, a +2.0 D converging lens adds about 2 dioptres of power to an eye that is already too strong. Distant rays are brought to a focus far in front of the retina, so the world blurs. Worse, the mismatch between the two eyes and the altered sense of distance impair judgement of steps and traffic - a genuine safety risk. Spectacles are not interchangeable between a converging defect and a diverging one.