Spherical Mirrors

Spherical Mirrors:

To derive the relevant formulae for reflection by spherical mirrors and refraction by spherical lenses, we must first adopt a sign convention for measuring distances. We shall follow the Cartesian sign convention.

According to this convention, all distances are measured from the pole of the mirror or the optical centre of the lens. The distances measured in the same direction as the incident light are taken as positive and those measured in the direction opposite to the direction of incident light are taken as negative .The heights measured upwards with respect to x-axis and normal to the principal axis (x-axis) of the mirror/ lens are taken as positive. The heights measured downwards are taken as negative.

There are different types of spherical mirrors, we use only two types in consideration namely concave and convex.

Focal Length of Spherical Mirrors:

The figure shows what happens when a parallel beam of light is incident on a concave mirror and a convex mirror. We assume that the rays are paraxial, i.e., they are incident at points close to the pole P of the mirror and make small angles with the principal axis. The reflected rays converge at a point F on the principal axis of a concave mirror. For a convex mirror, the reflected rays appear to diverge from a point F on its principal axis. The point F is called the principal focus of the mirror. If the parallel paraxial beam of light were incident, making some angle with the principal axis, the reflected rays would converge (or appear to diverge) from a point in a plane through F normal to the principal axis. This is called the focal plane of the mirror.

Concave mirror

Convex mirror

The focus point of these mirrors lie on a plane called the focal plane.

The distance between the focus F and the pole P of the mirror is called the focal length of the mirror, denoted by f. We now show that f = R/2, where R is the radius of curvature of the mirror. The geometry of reflection of an incident ray is shown.

Concave mirror

Convex mirror

Let C be the centre of curvature of the mirror. Consider a ray parallel to the principal axis striking the mirror at M. Then CM will be perpendicular to the mirror at M. Let be the angle of incidence, and MD be the perpendicular from M on the principal axis. Then,

∠MCP = and ∠MFP = 2

Now,

= and =

For small , which is true for paraxial rays, ≈ and ≈ 2 Therefore, the equation that has been derived changes as

= 2 or FD =

Now, for small , the point D is very close to the point P. Therefore, FD = and CD = R. Equation then gives

=

From this equation we can infer that the focal length of the mirror is equal to half the length of the radius of curvature.