Thin Lens Calculator
Inputs
| Focal length | 10 cm |
|---|---|
| Object distance | 30 cm |
Thin Lens Calculator
Find the image distance and magnification of a thin lens with the lens equation 1/f = 1/dₒ + 1/dᵢ. Enter the focal length and object distance to locate the image and see whether it is real or virtual, upright or inverted.
Inputs
Results
Enter a value to see results.
Thin Lens Calculator
A thin lens forms an image by bending light rays so they converge to — or appear to diverge from — a single point. The thin lens equation ties together the three distances that describe this: the focal length of the lens and the distances from the lens to the object and to its image. . "Thin" means the lens is treated as having no thickness, an excellent approximation for ordinary spectacle lenses, camera lenses and magnifying glasses.
This calculator takes the focal length and object distance and returns where the image forms, how large it is, and whether it is real or virtual.
Real and virtual images
The sign of the image distance tells you what kind of image you get. A positive image distance means a real image: the rays actually meet on the far side of the lens, so the image can be caught on a screen — this is how a projector or a camera works. A negative image distance means a virtual image: the rays only appear to come from a point on the same side as the object, so it cannot be projected but can be seen by looking through the lens, as with a magnifying glass.
Formula
| Quantity | Symbol | Meaning |
|---|---|---|
| Focal length | Lens focal length; positive for converging, negative for diverging | |
| Object distance | Distance from object to lens (always positive) | |
| Image distance | Distance from lens to image, | |
| Magnification | Height ratio, |
The magnification combines size and orientation in one number. Its magnitude is how many times larger or smaller the image is than the object; its sign tells you the orientation, with a negative value meaning the image is flipped upside down.
Worked example
A converging lens has a focal length of 10 cm. An object is placed 30 cm in front of it. The image distance is:
di=do−ff⋅do=30−1010×30=20300=15 cmThe magnification is . The positive image distance means a real image 15 cm behind the lens, and the negative magnification of 0.5 means it is inverted and half the size of the object — exactly what a camera does when it focuses a distant scene onto its sensor.
Converging and diverging lenses
A converging (convex) lens has a positive focal length and can produce either real or virtual images depending on where the object sits relative to the focal point. A diverging (concave) lens has a negative focal length and always produces a reduced, upright, virtual image, no matter where the object is. To model a diverging lens, enter a negative focal length. When the object sits exactly at the focal point, the outgoing rays are parallel and the image forms at infinity — the calculator flags this special case.
Limitations
The thin lens equation ignores the thickness of the lens and assumes paraxial rays — those that stay close to the central axis and make small angles with it. Real lenses suffer from aberrations: spherical aberration blurs rays far from the axis, and chromatic aberration spreads colours because the focal length depends slightly on wavelength. For precise optical design these effects must be modelled separately, but for everyday estimates the thin lens equation is remarkably accurate.
Frequently Asked Questions (FAQ)
What is the thin lens equation?
The thin lens equation relates the focal length f of a lens to the object distance dₒ and image distance dᵢ: 1/f = 1/dₒ + 1/dᵢ. Solving for the image distance gives dᵢ = f·dₒ / (dₒ − f). It assumes the lens is thin enough that its thickness can be ignored and that the light rays stay close to the central axis.
What sign convention does this use?
This calculator uses the standard convention for a single thin lens: the object distance dₒ is positive, a converging lens has a positive focal length and a diverging lens a negative one. A positive image distance means a real image on the far side of the lens; a negative image distance means a virtual image on the same side as the object. A negative magnification indicates an inverted image.
What does the magnification tell me?
The magnification m = −dᵢ/dₒ is the ratio of the image height to the object height. If |m| is greater than 1 the image is larger than the object; if it is less than 1 the image is smaller. The sign carries orientation: a negative m means the image is inverted (upside down), while a positive m means it is upright.
What is the difference between a converging and a diverging lens?
A converging (convex) lens is thicker in the middle and brings parallel rays to a focus, so it has a positive focal length and can form real images. A diverging (concave) lens is thinner in the middle and spreads parallel rays apart; it has a negative focal length and always forms a reduced, upright, virtual image. Enter a negative focal length to model a diverging lens.