A lens of +4 diopters is sold as a 2x magnifier. That is the example Wikipedia's article on magnifying glasses uses, and the label is honest in its own way: held right at the eye, with the page brought in to 12.50 cm, the lens does make print look twice as large. Put the page at the focal point instead, 25.00 cm away, relax your eye, and the same glass magnifies exactly 1.00x. Nothing about the lens changed. Only the question did.
Lenses are full of numbers like that, each one true and each one easy to read the wrong way. This article follows one plain, unmarked lens through three questions, what it is made of, where it puts the image and how much it magnifies, using the Refractive Index Calculator, the Thin Lens Calculator and the Magnifying Glass Calculator. Then come the five mistakes those numbers catch and the rules that avoid them. Every figure below came out of the calculators themselves.
A lens with nothing printed on it
Picture a round, clear lens from the back of a drawer. It is thicker in the middle than at the edge, which already says it gathers light rather than spreading it. There is no box, no marking and no record of what it was bought for. Three measurements, none of them needing more than a light source, a protractor, a lamp and a sheet of paper, are enough to find out what it is.
What is it made of?
The first clue is how strongly the material bends light. Suppose a refraction test on it gives two angles: light arriving at 60° from the normal bends to 35° inside. Snell's law turns that pair into a refractive index, n = 1.0003 × sin 60° / sin 35° = 1.5103, with 1.0003 standing for the air the light came from.
Inside the material, light moves at 198,496 km/s, or 123,340 mi/s, and every surface facing air reflects 4.13 % of the light that meets it head-on. What the calculator will not do is name the material with confidence. Two of its 18 reference materials fit: acrylic (PMMA) at 1.49 and crown glass N-BK7 at 1.5168.
The reason is the protractor. If either angle is off by half a degree, which is about as well as a hand-held protractor reads, the index could lie anywhere from 1.4842 to 1.5371. That window holds a cheap plastic magnifier and a good glass one alike. A second clue settles it better than a finer angle ever will: weight. Acrylic has a density of about 1.18 grams per cubic centimeter and N-BK7 2.51, so a glass lens is roughly twice as heavy as a plastic one of the same size, a difference you can check with the Density Calculator and a kitchen scale.
If you already have an index from a refractometer or a data sheet, the table shows what each commonly searched reading matches. Every value opens the calculator already set to it.
| Index reading | Matches in the calculator (within 1 %) |
|---|---|
| 1 | vacuum and air |
| 1.33 | water |
| 1.45 | fused silica |
| 1.48 | glycerol and acrylic (PMMA) |
| 1.49 | acrylic (PMMA) |
| 1.52 | crown glass (N-BK7) |
| 1.55 | rock salt (NaCl) and quartz |
| 1.6 | polycarbonate |
| 1.77 | sapphire and dense flint glass (N-SF11) |
| 2.16 | cubic zirconia |
| 2.42 | diamond |
| 2.65 | moissanite |
Three rows deserve a second look. 1.48 and 1.55 each fit two materials at once, and 1.77 cannot tell a sapphire from a dense flint glass. An index narrows the list. It rarely closes it.
Where does it put the image?
The second number is the focal length, and it can be measured without knowing the material at all. Set a lamp 30 cm in front of the lens and slide a sheet of paper behind it until the lamp appears sharp on the paper. Here that happens at 150 cm. The thin lens equation, 1/f = 1/30 + 1/150, gives a focal length of 25.00 cm, a converging lens of +4.00 D. The picture on the paper is real, upside down and 5.00 times larger than the lamp.
Now bring a page of print to 20 cm from the lens, inside that focal length. No image lands on paper anymore. The calculator puts it at -100.00 cm: the minus sign means it forms on the same side as the page, where it can be seen through the lens but never projected. It is upright and 5.00 times larger, so a letter 0.2 cm tall becomes a 1.00 cm image.
Move the page the other way and the picture changes character. At 25 cm, exactly the focal length, the light leaves as a parallel beam and the image runs off to infinity. At 40 cm it comes back as a real image 66.67 cm behind the lens, upside down and 1.67 times larger. The table inside the calculator walks the same lens through eight positions, from five focal lengths away to half of one.
How much does it magnify?
Here the numbers start to disagree with each other, and the disagreement is the point. The thin lens equation says the image of the page is 5.00 times larger. The Magnifying Glass Calculator, given the same +4 diopters and the same 20 cm, says the page looks 1.25x larger. Both are right, because they measure different things.
What the eye judges is not the size of an image but the angle it fills. The virtual image is five times bigger than the page, and it sits 100.00 cm away. Without a lens, the best view of the page is at the near point, the closest distance at which the eye can still focus, set by convention at 25 cm. An image five times larger at four times that distance looks 5 / 4 = 1.25 times larger. That is also 25 / 20, the near point divided by how far the page is from the lens, which is how the calculator works it out.
The lens has a range, not a single strength. With the page at the focal point, 25.00 cm, and a relaxed eye it gives 1.00x. Brought in to 12.50 cm, with the lens at the eye and the image at the near point, it reaches 2.00x. Anywhere between those two distances the view stays sharp. A detail 0.20 cm across looks like 0.20 cm at one end, 0.40 cm at the other and 0.25 cm at the 20 cm used here.
So the lens from the drawer is a +4 D magnifier, and a shop would sell it as 2x. Magnifiers and reading lenses are often marked in diopters, so here is the same arithmetic for the ratings people look up, with the focal length in inches. Each rating opens the calculator already set to it.
| Power | Focal length | Relaxed eye | Lens at the eye |
|---|---|---|---|
| +2 D | 19.69 in | 0.50x | 1.50x |
| +2.5 D | 15.75 in | 0.63x | 1.63x |
| +3 D | 13.12 in | 0.75x | 1.75x |
| +3.5 D | 11.25 in | 0.88x | 1.88x |
| +4 D | 9.84 in | 1.00x | 2.00x |
| +5 D | 7.87 in | 1.25x | 2.25x |
| +6 D | 6.56 in | 1.50x | 2.50x |
| +8 D | 4.92 in | 2.00x | 3.00x |
| +10 D | 3.94 in | 2.50x | 3.50x |
| +12 D | 3.28 in | 3.00x | 4.00x |
| +16 D | 2.46 in | 4.00x | 5.00x |
| +20 D | 1.97 in | 5.00x | 6.00x |
Below +4 D the relaxed-eye figure drops under 1x. A +2 D lens with the page at its focal point shows it at 0.50x, half the size it has at the near point. Lenses that weak only help held close to the eye, where they reach 1.50x.
Five mistakes the numbers catch
1. Reading the lens equation as what the eye sees. The 5.00 from the thin lens equation is linear magnification, the size of the image. The eye sees 1.25x, because that image is also four times farther away than the near point. Use the thin lens figure for projectors, cameras and anything else that throws an image onto a surface. Use the angular figure for anything you look through.
2. Trusting the X on the box. Labels are counted two ways, both starting from 250 mm. Counted as 250 mm / f + 1, a 2x label is the +4 D lens from this article, with a focal length of 25.00 cm. Counted as 250 mm / f, the same 2x is a +8 D lens with a focal length of 12.50 cm that gives 2.00x to 3.00x. Two boxes with the same number can hold lenses of twice the power. When a package also lists diopters, trust those.
3. Holding it at the wrong distance. The +4 D lens works between 12.50 cm and 25.00 cm from the page. Push the page out to 40 cm and it passes the focal length: the lens forms a real, upside-down image 66.67 cm away, which is why a magnifying glass held at arm's length turns the room on its head. Bring it in to 10 cm and the image sits only 16.67 cm from the lens, inside the near point, where the eye cannot focus on it and the print blurs.
4. Calling 1.5 glass. The round index often quoted for glass lands on acrylic in the calculator, because crown glass N-BK7 at 1.5168 sits 1.12 % above 1.5, just outside the 1 % window. The reading from the start of this article, 1.5103, fits both. Plastic and glass magnifiers overlap in index. They do not overlap in weight.
5. Assuming every eye gets the same magnification. Magnification compares the view through the lens with the best view without it, so it depends on the viewer's near point. The same +4 D lens gives 2.00x to 3.00x to an eye whose near point is 50 cm, and 4.00x to 5.00x to one at 100 cm. The glass is no stronger. The unaided view is worse, because that eye has to hold the page farther away to see it sharply. The near point moves outward with age, which is one reason the same lens can seem pointless to one reader and indispensable to another.
Rules worth keeping
- Focal length in centimeters is 100 divided by the diopters: +4 D is 25 cm and +10 D is 10 cm.
- Diopters divided by 4 give the relaxed-eye magnification. Add 1 for the lens held at the eye.
- Keep the page between the focal length and the distance that puts the image at your near point: for +4 D, 12.50 to 25.00 cm.
- A negative image distance means you look through the lens. A positive one means a screen can catch the image, upside down.
- Linear magnification sizes an image; angular magnification is what the eye sees. They agree only when the image sits exactly at your near point.
- A refractive index narrows down a material. Weight, hardness or a second test usually has to finish the job.
The lens from the drawer, in one table
| Question | What was measured | The answer |
|---|---|---|
| What is it made of? | 60° in, 35° out, from air | n = 1.5103, acrylic or crown glass |
| How sure is that? | either angle off by 0.5° | 1.4842 to 1.5371 |
| Where does it focus? | lamp at 30 cm, sharp at 150 cm | f = 25.00 cm, +4.00 D |
| Where is the image of a page 20 cm away? | thin lens equation | -100.00 cm, 5.00 times larger |
| How large does that page look? | near point of 25 cm | 1.25x |
| What range does the lens cover? | page from 25.00 to 12.50 cm | 1.00x to 2.00x |
| What would the box say? | 250 mm / f + 1 | 2x |
Tools discussed in this article
- Refractive Index Calculator: an index from two measured angles, a speed of light or a reading, matched against 18 reference materials at 589 nm
- Thin Lens Calculator: image distance, object distance or focal length in mm, cm or inches, with magnification, diopters and whether the image is real
- Magnifying Glass Calculator: what diopters or the X on a label really magnify, where to hold the page and how large print will look
More physics tools
The rest of the physics shelf, in the order it was built:
- Momentum Calculator, Force Calculator and Kinetic Energy Calculator
- Potential Energy Calculator, Work Calculator and Mechanical Power Calculator
- Torque Calculator, Kinetic Friction Calculator and Pendulum Period Calculator
- Thermal Expansion Calculator, Specific Heat Calculator and Thermal Conductivity Calculator
- Hooke's Law Calculator, Simple Harmonic Motion Calculator and Buoyancy Calculator
- Acceleration Formula Calculator and Temperature Converter
- Thermal Efficiency Calculator, Gravitational Force Calculator and Snell's Law Calculator, the last of them the single-surface version of everything a lens does twice