Curved mirrors form images by reflecting light rays in predictable ways, and the mirror equation connects the shape of the mirror to the position of the object and image. It is used for concave mirrors, which can focus light, and convex mirrors, which spread reflected rays outward. This relationship matters in telescopes, headlights, makeup mirrors, security mirrors, and many optical instruments.
By combining ray diagrams with equations, you can predict whether an image is real or virtual, enlarged or reduced, and upright or inverted.
The mirror equation is 1/f = 1/do + 1/di, where f is focal length, do is object distance, and di is image distance. Magnification is found with m = hi/ho = -di/do, which compares image height to object height and also shows image orientation. A concave mirror has a positive focal length and can make real or virtual images depending on object position, while a convex mirror has a negative focal length and always makes a virtual, upright, reduced image for a real object.
Careful sign conventions are essential because they determine whether distances, focal length, and magnification have the correct physical meaning.
Understanding Physics: The Mirror Equation
The mirror equation comes from a simple rule of reflection. Light hits a tiny part of the mirror surface, then leaves at the same angle on the other side of a line called the normal. For a spherical mirror, the normal points toward the center of curvature.
This geometry explains the useful rays in a ray diagram. A ray parallel to the main axis reflects through the focal point for a concave mirror. A ray aimed through the focal point reflects parallel to the axis.
A ray aimed at the center of curvature returns along its original path. Two reflected rays are enough to locate an image. More rays would meet at the same place in an ideal diagram.
The usual mirror calculations assume rays stay close to the main axis. These are called paraxial rays. Near the axis, the reflected rays from a spherical concave mirror come close to one common focus.
This is why the focal length is half the radius of curvature. Rays that strike near the mirror edge do not meet at exactly the same point. This effect is called spherical aberration.
It makes an image less sharp. A wide mirror opening collects more light, yet it can increase this blur. High quality telescopes often use specially shaped mirrors to reduce the problem.
Object position controls the kind of image formed by a concave mirror. An object far beyond the center of curvature produces a smaller image between the center and the focus. As the object moves toward the center, the image grows.
At the center, image size matches object size. Between the center and the focus, the image forms farther from the mirror and becomes enlarged. When the object is inside the focal length, reflected rays spread apart.
Your eye traces those rays backward, so the image appears behind the mirror. A screen cannot catch this image because no light actually gathers there. This is why a makeup mirror can show an upright enlarged face, while a projector needs a real image on a screen.
Sign conventions are not just a rule to memorize. They keep the calculation connected to the path of light. Start every distance at the mirror surface, often called the pole.
Use one unit throughout, such as centimetres or metres. A negative image distance tells you to look behind the mirror in the diagram. A negative magnification tells you the image is inverted.
Convex vehicle mirrors provide a useful check. They form images behind the mirror, reduce their size, and give a wider field of view.
When solving a problem, sketch the axis first, mark the focus and center, then estimate the expected image before calculating. This catches many errors in signs, units, and calculator input.
Key Facts
- Mirror equation: 1/f = 1/do + 1/di.
- Magnification equation: m = hi/ho = -di/do.
- For the standard sign convention, concave mirrors have f > 0 and convex mirrors have f < 0.
- A real image has di > 0 and forms in front of the mirror where reflected rays actually meet.
- A virtual image has di < 0 and appears behind the mirror where reflected rays only seem to originate.
- The center of curvature is at C = 2f, so the radius of curvature satisfies R = 2f.
Vocabulary
- Focal length
- The distance from the mirror to the focal point, where rays parallel to the principal axis meet or appear to diverge from after reflection.
- Object distance
- The distance do from the object to the mirror, measured along the principal axis.
- Image distance
- The distance di from the image to the mirror, with its sign showing whether the image is real or virtual.
- Magnification
- The ratio m = hi/ho that tells how large the image is compared with the object and whether it is upright or inverted.
- Principal axis
- The straight reference line through the mirror's center that also passes through the focal point and center of curvature.
Common Mistakes to Avoid
- Using the lens sign convention for mirrors, because mirror problems have their own rules for positive and negative image distance and focal length.
- Forgetting that convex mirrors have negative focal length, because this changes the calculated image distance and magnification.
- Calling every image in front of a mirror virtual, because a real image forms in front of a mirror when reflected rays actually converge there.
- Ignoring the negative sign in m = -di/do, because the sign of magnification tells whether the image is upright or inverted.
Practice Questions
- 1 A concave mirror has focal length f = 12 cm. An object is placed do = 36 cm in front of the mirror. Use 1/f = 1/do + 1/di to find the image distance and state whether the image is real or virtual.
- 2 A convex mirror has focal length f = -20 cm. An object 6 cm tall is placed do = 30 cm in front of it. Find di, the magnification m, and the image height hi.
- 3 A student moves an object from beyond the center of curvature to between the focal point and a concave mirror. Describe how the image changes in position, size, orientation, and whether it is real or virtual.