Spherical mirrors are generally constructed from glass. One surface of the glass is silvered. The reflection takes place at the other surface. If reflection takes place at the convex surface (figure 3b), it is called a convex mirror and if reflection takes place at the concave surface (figure 3a), it is called a concave mirror. Some common examples of concave mirrors are shaving mirrors and makeup mirrors. These types of mirrors magnify objects placed close to them. A common example of convex mirrors is the passenger-side wing mirrors of cars. These type of mirrors have wider fields of view than equivalent flat mirrors, but objects which appear in them generally look smaller (and, therefore, farther away) than they actually are.
Some definitions regarding spherical mirrors
- The centre of the sphere, of which the mirror is a part, is called the centre of curvature of the mirror (Point C in figure 3).
- The radius of this sphere is called the radius of curvature of the mirror (length of CP in figure 3).
- The line joining the pole and the centre of curvature is called the principal axis (Straight line PC in figure 3).
- Suppose a light beam travelling in a direction parallel to the principal axis is incident on a concave mirror. The point where the reflected rays converge is called the focus of the mirror (Point F in figure 3a). In case of a convex mirror, the reflected rays appear to diverge from a point on the principal axis (Point F in figure 3b). This point is the focus of the convex mirror.
- The plane through the focus and perpendicular to the principal axis is called the focal plane.
- The distance of the focus from the pole is called the focal length of the mirror (length of FP in figure 3).
- A ray close to the principal axis is called a paraxial ray.
Real and virtual images
When a point object is placed before a spherical mirror of small aperture, a point image is formed. A reflected ray is traced by applying the laws of reflection. If the reflected rays intersect, the point of intersection is the real image. If the rays diverge after reflection, a virtual image is formed at the point from where the rays seem to diverge. Figure 4 shows some examples.
In the sign convention for spherical mirrors, the pole1 is taken to be the origin and the principal axis as the x-axis. The quantities u, v, R and f denote the x-coordinates of the object, the image, the centre of curvature and the focus respectively. The sign convention for spherical mirrors is summarised in table 1.
It can be proved that the quantities u, v and R are related through the relation
Extended Objects and Magnification
Suppose an object AB is placed on the principal axis of a spherical mirror with the length AB perpendicular to the principal axis (figure 5a). Consider two rays BD and BE, the first parallel to the principal axis and the other directed towards the centre of curvature. The image of B is formed at the intersection of these two reflected rays. Thus, A’B’ is the image of AB. The lateral or transverse magnification is defined as
where h2 = height of the image and h1 = height of the object. The height of the object (placed perpendicular to the principal axis) is taken to be positive. If the image is also on the same side of the principal axis, its height is also positive. The image is then erect. If the image is inverted, its height is taken as negative. Figure 5b shows an example of inverted image. It can be shown that the lateral magnification of an extended object by a spherical mirror
1. This video shows reflection of a parallel light beam from a concave mirror –
2. This video shows reflection of a parallel light beam from a convex mirror –
1. This website illustrates different cases of image formation by spherical mirrors: http://dev.physicslab.org/Document.aspx?doctype=3&filename=GeometricOptics_SphericalMirrors.xml
1 The point on the mirror at the middle of the surface is called its pole (Point P in figure 3).
MCQs on SPHERICAL MIRRORS
A point object is placed at a distance of 30 cm from a convex mirror of focal length 30 cm. The image will form at
- 15 cm behind the mirror.
The image formed by a convex mirror of a real object is
- virtual and erect
- virtual and inverted
- real and erect
- real and inverted
A virtual image larger than the object can be produced by
- concave mirror
- convex mirror
- plane mirror
- concave lens
In image formation from spherical mirrors, only paraxial rays are considered because they
- are easy to handle geometrically
- contain most of the intensity of the incident light
- form nearly a point image of a point source
- show minimum dispersion effect.
Which of the following (referred to a spherical mirror) depends on whether the rays are paraxial or not?
- Radius of curvature
- Principal axis
Question 01: D
Question 02: A
Question 03: A
Question 04: C
Question 05: B