Concepts: reflection by spherical mirrors: Ray Optics And Optical Instruments

2.1 Reflection by spherical mirrors

A spherical mirror is a part of a sphere whose inner or outer surface is polished. The centre of curvature is the centre of that sphere, the radius of curvature is its radius, and the pole is the centre of the mirror surface. The principal axis is the line through the pole and the centre of curvature.

The reflection of a ray at the mirror obeys two simple rules. A ray parallel to the principal axis passes, after reflection, through the focus. A ray through the centre of curvature comes back along the same path. The focus is midway between the pole and the centre of curvature, so the focal length is half the radius of curvature.

The Cartesian sign convention removes the guesswork from the formulas. Measure every distance from the pole, take the direction of the incident light as positive, and give heights above the principal axis a positive sign. A concave mirror has its centre of curvature on the side of the incident light, so for it u is negative and f is negative. A convex mirror has the centre behind the mirror, so f is positive.

2.1.1 Image formation

The image position follows from the mirror formula once the signs are fixed. For a concave mirror, an object beyond the centre gives a real inverted diminished image, an object between the centre and the focus gives a real inverted magnified image, an object at the focus produces an image at infinity, and an object between the pole and the focus gives a virtual erect magnified image. A convex mirror always gives a virtual erect diminished image, which is why it is used as a rear-view mirror.

2.1.2 Uses and the mirror equation

Concave mirrors are used in headlights, searchlights and dentist's mirrors because they can concentrate light or magnify an object placed close to them. Convex mirrors are used in vehicles because their wide field of view keeps a smaller virtual image.

The mirror equation is written as one over v plus one over u equals one over f. The magnification is the ratio of the image height to the object height, which also equals minus v over u. A negative magnification means the image is inverted relative to the object.

2.1.3 Exercise coverage

The exercises on this section ask for image positions, nature and size of images, and the relation between the radius of curvature and the focal length. A typical question gives the object distance and the radius of curvature, then asks for the screen position and the magnification. Write the sign convention step, substitute, and interpret the negative sign as a real inverted image.