How to use the thin lens calculator
- Choose what to find: the image distance, the object distance or the focal length.
- Enter the other two values in any one length unit. Use a positive focal length for a converging (convex) lens and a negative one for a diverging (concave) lens.
- Read the missing distance, the magnification, whether the image is real or virtual, upright or inverted, and enlarged or reduced.
- Add the object height in the advanced options to get the height of the image.
Formula
1/f = 1/do + 1/di
magnification m = -di / do
image height = m x object height
Newton form: (do - f) x (di - f) = f^2
do is the distance from the object to the lens and di the distance from the lens to the image. A positive di means a real image on the far side of the lens. A negative di means a virtual image on the object side, and a negative m means the image is inverted.
A 50 mm lens and a magnifier
A converging lens with f = 50 mm and an object 200 mm away has 1/di = 1/50 - 1/200, so di = 66.67 mm and m = -0.3333. The image is real, inverted and one third the size. An object 10 mm tall gives an image 3.33 mm tall. The Newton check agrees: (200 - 50) x (66.67 - 50) = 2500 = 50 x 50.
A magnifier with f = 100 mm and an object 60 mm away gives di = -150 mm and m = 2.5. The negative sign means a virtual image on the object side, upright and enlarged 2.5 times, which is why you see a bigger picture when you hold a magnifier close to a page. For a diverging lens of f = -50 mm and an object 100 mm away, di = -33.33 mm.
Real and virtual images
A real image is formed where rays actually meet, so a screen placed there shows the picture. This is how a camera lens and a projector work. A virtual image is where the rays only appear to come from. You see it by looking through the lens, as with a magnifier or glasses. A converging lens makes a virtual image only when the object is closer than the focal length.
Choosing the unknown
Cameras and projectors usually fix the focal length and ask for the image distance. A designer who knows where the object and image must be solves for the focal length. A tester who has measured the focal length and the image distance can recover the object distance. Link the result to the Telescope Magnification Calculator to see how two lenses combine.
More waves and light tools
For related tools on light and sound, try the Telescope Magnification Calculator, Doppler Effect Calculator and Speed of Sound Calculator.
Assumptions and limits
- The lens is thin, so its thickness is ignored.
- Light rays are close to the axis of the lens.
- There is a single lens in air, and a real object (a positive object distance).
- All distances use one length unit.
- The focal length and distances in the form are examples to replace with your own.
Frequently asked questions
What happens when the object is at the focal length?
The rays leave the lens parallel and never meet, so the image is infinitely far away. The tool refuses this case and says why instead of showing a meaningless number.
Why is the magnification negative?
The sign tells you the orientation. A negative value means the image is upside down compared with the object, as with a camera image. A positive value means it is upright.
Can I use this for a camera lens that has several elements?
Only as a rough guide. A real camera lens has many elements and thick glass, so use the focal length and distances given by the maker as the effective values.
Why can the image distance be negative?
A negative image distance means the image is virtual and on the same side of the lens as the object. It happens with a diverging lens or a converging lens used as a magnifier.