Reflection: Spherical Mirrors

A spherical mirror is a small part of a hollow sphere whose one side is silvered. If the inner (hollow) side reflects light it is a concave mirror; if the outer (bulging) side reflects, it is a convex mirror. A concave mirror converges parallel light to a real focus in front of it, so it is a converging mirror. A convex mirror diverges parallel light, which only appears to come from a focus behind it, so it is a diverging mirror. The line through the pole and the centre of curvature is the principal axis, the reference line for everything we draw.

Four points matter. The pole P is the middle of the mirror. The centre of curvature C is the centre of the sphere the mirror was cut from, and the distance PC is the radius of curvature R. The focus F is where rays parallel to the principal axis converge after reflection (concave) or seem to diverge from (convex). The distance PF is the focal length f. For a spherical mirror the focus sits exactly halfway, so f = R/2. Knowing these points lets us predict every image by drawing just two rays.

To locate an image, draw any two of these standard rays from the top of the object. A ray parallel to the principal axis reflects through the focus F. A ray passing through the focus F reflects back parallel to the axis. A ray through the centre of curvature C hits the mirror normally and returns along its own path. Where the reflected rays meet is the top of the image. If they actually cross in front of the mirror the image is real; if only their backward extensions meet behind the mirror, the image is virtual.

For a concave mirror the image changes as the object moves: far away it is small, real and inverted; at C it is the same size; inside the focus it becomes large, erect and virtual (the make-up or shaving mirror). A convex mirror always gives a small, erect, virtual image, so it shows a wide view and is used as a vehicle rear-view mirror. Numbers come from the mirror formula 1/v + 1/u = 1/f, with distances measured by the New Cartesian sign convention. Magnification m = -v/u tells the size and nature: positive m means erect (virtual), negative m means inverted (real).