Convex and Concave Lenses: Ray Diagrams
A convex (converging) lens is thicker at the center and brings parallel rays to a focus at the focal point, while a concave (diverging) lens is thinner at the center and spreads parallel rays so that they appear to come from a virtual focal point; ray diagrams use three special rays to locate the image. In brief: a convex lens can form real, inverted images (object beyond the focal length) or a virtual, upright, magnified image (object inside the focal length, as in a magnifying glass). A concave lens always forms a virtual, upright, diminished image. The focal length f is the distance from the lens to the focal point.
Lenses in eyeglasses, cameras, microscopes and magnifying glasses form images by refraction. This guide explains the shapes and terms for lenses, shows how to draw ray diagrams with the three principal rays, compares the image types in a table, works through the cases for a lens with f = 20 cm and corrects common mistakes.
What are convex and concave lenses?
A lens is a piece of transparent material, usually glass or plastic, with one or two curved surfaces. Light is refracted at each surface. A convex lens (converging, positive) bends rays toward the principal axis. Parallel rays from a distant object converge at the principal focus F, at a distance f (the focal length) from the lens. A concave lens (diverging, negative) bends rays outward, so the rays appear to diverge from a virtual focus on the incoming side. Each lens has two focal points, one on each side, at the same distance. The line through the center of the lens, perpendicular to it, is the principal axis. A thicker or more curved lens (made of a material with a higher refractive index) has a shorter focal length and is more powerful.
What are the three principal rays?
To find the image of the top of an object, draw at least two of these rays from the top of the object (a thin lens is shown by a line with arrows):
- Ray parallel to the axis: after the lens, it passes through the focal point on the far side (convex lens), or appears to come from the focal point on the near side (concave lens).
- Ray through the center of the lens: it continues in a straight line without deviation.
- Ray through the near focal point (convex lens; aimed at the far focal point for a concave lens): after the lens, it travels parallel to the axis.
The image of the top of the object is where the rays meet (a real image, which can be projected on a screen) or where their backward extensions meet (a virtual image, seen only by looking through the lens). The image is described by its position, size (magnification), orientation (upright or inverted) and type (real or virtual).
Image characteristics compared
| Object position | Image position | Size | Type and orientation | Example |
|---|---|---|---|---|
| Convex: beyond 2f | Between f and 2f | Diminished | Real, inverted | Camera |
| Convex: at 2f | At 2f | Same size | Real, inverted | Photocopier |
| Convex: between f and 2f | Beyond 2f | Magnified | Real, inverted | Projector |
| Convex: at f | At infinity | — | Rays emerge parallel | Searchlight |
| Convex: inside f | Same side as the object | Magnified | Virtual, upright | Magnifying glass |
| Concave: any position | Between lens and focal point, same side | Diminished | Virtual, upright | Peephole, myopia glasses |
Worked example: images for different object distances
A convex lens has a focal length of f = 20 cm. The thin-lens equation 1/f = 1/do + 1/di (explained in the next article) gives the image distance di and the magnification m = −di ÷ do.
- Object at 50 cm (beyond 2f = 40 cm): 1/di = 1/20 − 1/50 = 0.050 − 0.020 = 0.030, so di = 33 cm. m = −33 ÷ 50 = −0.67: real, inverted, diminished.
- Object at 40 cm (= 2f): di = 40 cm, m = −1: real, inverted, the same size.
- Object at 30 cm (between f and 2f): 1/di = 0.050 − 0.0333 = 0.0167, di = 60 cm. m = −2.0: real, inverted, twice the size.
- Object at 20 cm (at f): 1/di = 0, so the image is at infinity.
- Object at 10 cm (inside f): 1/di = 0.050 − 0.100 = −0.050, di = −20 cm. The negative sign means a virtual image on the same side as the object; m = +2.0: upright, twice the size.
- Concave lens, f = −15 cm, object at 45 cm: 1/di = −1/15 − 1/45 = −0.0667 − 0.0222 = −0.0889, di = −11 cm, m = +0.25: virtual, upright, one quarter of the size.
How were lenses developed?
The Nimrud lens, a piece of polished rock crystal from Assyria (about 700 BCE), may have been a magnifier. Eyeglasses were invented in Italy around 1286. The telescope and the microscope were developed in the Netherlands in the early 1600s, and Galileo’s improvements of the telescope in 1609 led to the discovery of the moons of Jupiter. Johannes Kepler explained the ray paths in lenses in his Dioptrice (1611).
Which mistakes are common?
Mistake 1: Drawing the lens as if it bent the rays in the middle
In a thin-lens diagram, the rays are bent at the central line of the lens.
Mistake 2: Saying that all images from a convex lens are inverted
Inside the focal length, the image is upright and virtual.
Mistake 3: Forgetting that virtual images cannot be put on a screen
Only real images can.
Mistake 4: Confusing the focal point with the center of curvature
For thin lenses, the focal length depends on the curvatures and on the refractive index, not just on the radius of the surface.