REFLECTION-AT SPHERICAL MIRRORS

A spherical mirror is a reflecting surface which forms a part of a sphere (as shown in following a and b diagram). When the reflection takes place from the inner surface and outer surface is polished or silvered the mirror is known as concave mirror. Vice- versa, it is convex.

                                       Ray diagram for concave and convex mirror

CHARACTERISTICS OF REFLECTION BY A SPHERICAL REFLECTING SURFACES

(FOR SMALL APERTURE)
(i)    Pole (P) is generally taken as the mid point of reflecting surface.
(ii)    Centre of curvature (C) is the centre of the sphere of which the mirror is a part.
(iii)    Radius of curvature is the radius of the sphere of which the mirror is a part. Distance between P and C.
(iv)    Principal Axis is the straight line connecting pole P and centre of curvature C.
(v)    Principal focus (F) is the point of intersection of all the reflected rays which strike the mirror (with small aperture) parallel to the principal axis. In concave mirror it is real and in the convex mirror it is virtual.
(vi)    Focal length (f) is the distance from pole to focus.
(vii)    Aperture is the diameter of the mirror.
(viii)    If the incident ray is parallel to the principal axis, the reflected ray passes through the focus. (Fig (a))

 (ix)    If the incident ray passes through the focus, then the reflected ray is parallel to the principal axis (Fig.(b))

(x)    Incident ray passing through centre of curvature will be reflected back through the centre of curvature. (Fig.(c))

(xi)     F = R/2,  where f = focal length R = Radius of curvature

(xii)    Sign convention and magnification. We follow Cartesian coordinate system convention according to which:
          (a)    The pole of the mirror is the origin 
          (b)    The direction of the incident rays is considered as positive x-axis.
          (c)    Vertically up is positive y-axis
        Note: Radius of Curvature and Focal Length of:
          (a)    Concave mirror is positive
          (b)    Convex mirror is positive
(xiii)   (a)    Linear Magnification or lateral magnification or transverse magnification 

                                       = M = h₂ / h₁ = - v/u

 where h₂ = y  coordinate of image;  h_{1 =} y coordinate of the object

                 (both perpendicular to the principle axis of mirror)

          (b)    Longitudinal Magnification  (for any size of the object)  (for short object)

Tracing for spherical mirror

(1)    A ray going through centre of curvature is reflected back along the same direction.     
 
(2)    A ray parallel to principal axis is reflected through the focus, and vice-versa. Also, mutually parallel rays after reflection intersect on the focal plane.     
 
(3)    A ray going to the pole and the reflected ray from it make equal angles with the principal axis.