Converging and Diverging Lenses: Locating Images by Ray Diagrams

1A picture from one piece of glass

Stand at the back of a classroom, facing a sunny window. Hold a magnifying glass between the window and a sheet of white paper, and slowly move the paper back and forth. At one distance a sharp, bright picture of the window appears on the paper, with the trees and sky outside it, all upside down. One curved piece of glass has made a picture you can catch on a screen. A camera, a projector and your own eye all work this way. This lesson explains how, by drawing the paths of a few rays.

2Words for describing a lens

• The optical centre O is the centre of the lens. A ray through O passes straight through without being deviated.
• The principal axis is the line through O at right angles to the lens.
• The principal focus F is the point on the principal axis where rays parallel to the axis meet after passing through a converging lens, or from which they appear to spread out after a diverging lens. A lens has a principal focus on each side, at the same distance.
• The focal length $ff$ is the distance from the optical centre to the principal focus. A fatter, more curved lens bends light more and has a shorter focal length.
• The point labelled 2F is at twice the focal length from the lens. It is a useful landmark in ray diagrams.

The focus of a converging lens is real: light really passes through it, and you could catch a bright spot there on paper. The focus of a diverging lens is virtual: the rays only seem to come from it.

3Experiment: a quick focal length

Repeat the window experiment with a metre rule. When the picture of a distant tree or building is sharpest, measure the distance from the lens to the paper. Light from a very distant object arrives almost parallel, so it is focused at F, and this distance is the focal length. For a lens with $f=10\mathrm{c}\mathrm{m}f\; =\; 10\backslash ,\backslash mathrm\{cm\}$, an object 5 m away gives a sharp image at 10.2 cm, so the estimate is within about 2%.

4Three rays that are easy to draw

Light from the tip of an object spreads out in all directions, but three rays have paths you can predict without measuring any angles. For a ray diagram, the lens is drawn as a vertical line on the principal axis, with arrowheads pointing out for converging and in for diverging.

Two rays are enough to locate the image; the third is a check. Where the rays meet after the lens, the image is real. If they spread out after the lens, extend them backwards with dashed lines: where the dashed lines meet is a virtual image.

Drawing a ray diagram to scale

1. Choose a scale, for example 1 cm on paper for 5 cm in reality, and write it down.
2. Draw the principal axis and the lens. Mark F and 2F on both sides.
3. Draw the object as an upright arrow at the correct distance.
4. Draw at least two of the three rays from the tip of the object, with arrows to show direction.
5. Draw the image from the axis to where the rays meet. Measure its distance and height, and convert with your scale.

5Where is the image? Following the object in

Each diagram below was drawn to scale from the lens equation of lesson 4, so the rays really do meet where the image is shown. Ray 1 is red, ray 2 green and ray 3 blue.

Object beyond 2F: the camera and the eye

The image is real, inverted, diminished, and lies between F and 2F on the other side. A camera lens and the lens of your eye form images like this on the sensor or the retina.

Object between F and 2F: the projector

The image is real, inverted, magnified, and lies beyond 2F. A projector puts a small, brightly lit slide just outside F, upside down, so that a large upright picture appears on a distant screen.

Object between F and the lens: the magnifying glass

The rays spread out after the lens and never meet. Your eye traces them back to a virtual, upright, magnified image on the same side as the object. This is how a magnifying glass works: hold it so that the object is closer than one focal length.

The diverging lens

Wherever the object is, a diverging lens gives a virtual, upright, diminished image between F and the lens, on the same side as the object. The lenses in spectacles for short sight and the wide-angle viewer in some doors are diverging.