You’ve been hired by a community vision clinic. Three clients are waiting, and each can’t see something they need to see. The clinic has a limited stock of lenses and a tight budget for each client. Your job is to design the right aid for one of them: choose the lens, prove with the physics that it meets their need and every constraint, and then explain it to them in plain language, including what it can’t do.
🧰Your Toolkit
Everything you need is in this unit. Review the lens activity first if the equation feels shaky.
Pick one case. The eye data and constraints are everything the clinic knows. Assume nothing beyond what is written.
Case
Client and need
Eye data
Constraint
A
Mr. Alvarez, 74. He can’t read his pill bottle. He needs a hand-held magnifier.
Reads print 3.0 mm tall comfortably at 25 cm. The label print is 1.2 mm tall.
Lens at least 4.0 cm across, so he can see a whole word, and at most $12. Pick from the catalog.
B
Priya, 15. She can’t read the board. She needs distance glasses, and still has to read a book at 25 cm.
Nearsighted. She sees clearly only between 0.10 m (near point) and 0.45 m (far point).
The pair costs at most $80. Lens powers come in 0.25 D steps. Don’t over-correct by more than 0.50 D.
C
Mr. Okafor, 58. He has to hold a book at arm’s length to read it. He needs reading glasses.
Farsighted. His near point is 0.80 m. Distance vision is fine.
The pair costs at most $80. Lens powers come in 0.25 D steps. He must be able to read a book held anywhere from 25 to 30 cm away.
Case A catalog:
Catalog lens (Case A)
Focal length f
Power
Diameter
Price
L1
12.5 cm
+8.0 D
7.5 cm
$6
L2
10.0 cm
+10.0 D
6.0 cm
$7
L3
8.0 cm
+12.5 D
5.0 cm
$8
L4
6.0 cm
+16.7 D
4.0 cm
$10
L5
5.0 cm
+20.0 D
3.0 cm
$12
L6
4.0 cm
+25.0 D
2.5 cm
$15
L7
3.0 cm
+33.3 D
2.0 cm
$20
Cases B and C prices: each lens costs $28 plus $4 for every diopter of power (positive or negative), so a pair of lenses costs 2 × ($28 + $4 × |P|).
🧮The Method
A corrective lens doesn’t fix the eye. It takes the thing the client is looking at and puts a virtual image where their eye can already focus: at the far point for distance vision, or at the near point for reading. Use the lens equation with a negative image distance, with every distance in meters so that 1/f is the power in diopters.
Lens equation:P = 1/f = 1/d_o + 1/d_i (d_i is negative for a virtual image) Distance vision (nearsighted): object at ∞, image at the far point, so P = −1 ÷ far point Reading (farsighted): object at d_o, image at the near point, so P = 1/d_o − 1/near point Check any use:1/d_i = P − 1/d_o, and the image must land between the near point and the far point Reading window (converging lens): from d_o = 1 ÷ (P + 1/near point) out to d_o = f = 1/P Magnifier:M ≈ 25 cm ÷ f. The print looks M times as tall, and the working distance is about f
Worked example 1: distance glasses (a different client)
Given
Ms. Kim is nearsighted, with a far point of 0.80 m and a near point of 0.10 m
Solve
Distant object, so d_o = ∞ and 1/d_o = 0
The image must be virtual at her far point: d_i = −0.80 m
P = 1/d_o + 1/d_i = 0 + 1/(−0.80) = −1.25 D
Second use, a book at 25 cm: 1/d_i = −1.25 − 4.00 = −5.25 m⁻¹, so d_i = −0.19 m
The image is 19 cm away, inside her range (10 cm to 80 cm). She can still read.
✓ A −1.25 D diverging lens fixes distance vision and still lets her read at 25 cm.
Worked example 2: reading glasses (a different client)
Given
Mr. Lee is farsighted, with a near point of 1.20 m, and wants to read at 25 cm
Solve
d_o = 0.25 m, and the image must be virtual at his near point: d_i = −1.20 m
P = 1/0.25 + 1/(−1.20) = 4.00 − 0.83 = +3.17 D
Nearest stock step: +3.25 D
Reading window with +3.25 D: from 1 ÷ (3.25 + 0.83) = 24.5 cm out to f = 1 ÷ 3.25 = 30.8 cm
✓ A +3.25 D converging lens works for reading at 25 cm. It lets him read anywhere from 24.5 to 30.8 cm.
📏Constraints
Pick one client file, or use the one your teacher assigns.
Your design has to meet the client’s need and every constraint in their file, and you have to show it with calculations.
Use the lens equation and the eye data in the file. Show every step with units.
Check your lens against a second use, not just the main need. A lens that fixes one problem can create another.
Explain your design to the client in plain language, including why a weaker or stronger lens wouldn’t be as good.
Name one tradeoff you made, and one limit of your design.
🚚Deliverables
A design: the lens (a catalog lens for Case A, or a power in diopters for Cases B and C), with the calculation that leads to it.
A verification: every constraint in the client’s file checked with a number. The Designer can confirm your work.
A client explanation of four to six sentences, in plain language.
A short note on one tradeoff you made and one limit of your design.
🗓️Suggested Pacing (one class day)
First 5 min: Pick. Choose a client and read the whole file.
Next 15 min: Calculate. Work out the lens by hand, with every step.
Next 10 min: Check. Confirm every constraint in the Designer, and test a second use.
Next 15 min: Explain. Write the client explanation and the tradeoff and limit.
Last 5 min: Review. Reread against the rubric.
👓Design the Aid
Calculate first, then use the Designer to confirm. If a check fails, work out why before you change the lens.
👓 Vision Aid Designer
Choose your client’s case, then try a lens. Work out your choice by hand first, then use this to check it. Each check turns green when the lens meets that part of the client’s file.
Client:
Board (object at ∞): 1/d_i = P, so the image is at ∞ (no correction) Book at 25 cm: 1/d_i = P − 1/0.25 = -4.00 m⁻¹, so the image is 25.0 cm away
✗ Distant objects: the image lands at ∞ away (must be 45 cm or closer, her far point)
✗ Over-correction: none (needs a negative power) against the exact value (limit 0.50 D)
✓ Reading at 25 cm: the image lands 25.0 cm away (must be between 10 and 45 cm)
✓ Cost of the pair $56 (limit $80)
Not there yet. Adjust the lens.
✍️Client Explanation
Write four to six sentences to your client, in plain language, with no equations. Cover:
What the problem is. Say where the client’s eye focuses, or how big the print looks, in everyday terms.
What your lens does. Explain that it moves the image to where the eye can focus, or makes the print bigger.
Why this one. Say why a weaker or a stronger lens wouldn’t be as good.
What to expect. Mention the other situation you checked, such as reading, distance, or field of view.
Tradeoff and limit. One thing you gave up, and one thing the design can’t do.
📋Rubric
Criterion
Weight
What we’re looking for
Physics & Calculations
35%
Uses the lens equation, signs, and power correctly with the client’s eye data, and shows every step with units.
Meets Need & Constraints
25%
The design passes every constraint in the file, and each one is checked with a number.
Second-Use Check
15%
Checks the lens in another situation (reading for B, a second distance for C, field of view and working distance for A) and says what it finds.
Explanation & Tradeoffs
20%
Explains in plain language why the lens works and why a different one wouldn’t, and names a real tradeoff and a real limit.
Communication
5%
Clear enough that the client, or a classmate, could follow it.
📤 Turn In
Your design and the calculation behind it, with every step and unit
Your verification of each constraint in the client’s file, with numbers