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Lenses and Mirrors
Waves & Optics
Lenses bend light to form images, and mirrors reflect it to do the same. A few simple equations predict where the image will be and how big it will appear.
Lenses
A lens is a transparent object, usually glass or plastic, with curved surfaces that refract light. A converging lens is thicker in the middle and brings parallel rays to a focus at its focal point, a distance f from the lens. A diverging lens is thinner in the middle and spreads parallel rays apart, so they appear to come from a virtual focal point on the same side. The thin lens equation relates focal length, object distance and image distance, and magnification tells you how large the image is compared with the object. A positive magnification means upright, a negative one means inverted.
Mirrors
A mirror forms an image by reflection. A plane mirror gives a virtual, upright image the same size as the object. A concave mirror curves inward and can converge rays to a real focus, which is why it is used in shaving mirrors and headlights. A convex mirror curves outward, spreads rays apart, and always gives a smaller virtual image, which is why it is used as a wide-angle mirror on cars and in shops. Spherical mirrors obey the same form of equation as lenses, 1/f = 1/do + 1/di, with the focal length equal to half the radius of curvature, but the sign conventions are different.
Where you see it
Your eye is a camera: its lens focuses light onto the retina, and glasses correct the focus when the eye is too long or too short. Cameras, microscopes and telescopes all use combinations of lenses to magnify small or distant objects. A telescope's large objective lens or mirror gathers light, and its eyepiece magnifies the image. Contact lenses, magnifying glasses, projectors and the lenses in a phone camera all work from the same two equations. Even a drop of water can act as a lens, which is why a water droplet on a leaf can magnify the veins beneath it.
Worked example: image from a converging lens
A converging lens has focal length f = 10 cm. An object is placed do = 30 cm from the lens. Use 1/f = 1/do + 1/di. Substitute: 1/10 = 1/30 + 1/di. So 1/di = 1/10 - 1/30 = 3/30 - 1/30 = 2/30 = 1/15. Therefore di = 15 cm. Magnification m = -di/do = -15/30 = -0.50. The image is real, inverted, 15 cm past the lens, and half the size of the object.
Common mistakes
- Mixing up the sign convention: for a lens, a positive image distance means a real image on the far side; a negative magnification means the image is inverted.
- Using the lens equation for a mirror without care: the equation has the same form, but the sign rules for mirrors differ from those for lenses.
- Thinking a diverging lens makes a real image: a single diverging lens always forms a virtual, upright, smaller image.
Practice questions
- An object is 40 cm from a lens of focal length 20 cm. Where is the image and what is the magnification?
Show answer
1/di = 1/20 - 1/40 = 1/40, so di = 40 cm; m = -40/40 = -1.0, so the image is the same size, inverted.
- Which type of lens corrects short sight?
Show answer
A diverging lens, with a negative focal length.
- What kind of image does a plane mirror form?
Show answer
It is virtual and upright, as far behind the mirror as the object is in front.
Frequently asked questions
What is the difference between a real and a virtual image?
A real image can be projected onto a screen; a virtual image cannot, because the rays only appear to come from it, as in a plane mirror.
How do glasses correct short sight?
Use a diverging lens, which spreads the rays so the image forms on the retina instead of in front of it.