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Formation and Laws of Reflection of Light at Spherical Mirrors
So far we have discussed the reflection of light from plane surfaces like that of a plane mirror. When a parallel beam of light rays falls on a plane mirror, it is reflected as a parallel beam. So, a plane mirror changes only the direction of incident light rays, it does not ‘converge’ or ‘diverge’ the parallel rays of light (To bring the parallel rays of light ‘closer together’ is called ‘to converge’ the light rays whereas ‘to spread out’ the parallel rays of light is called ‘to diverge’ the light rays).
We will now describe the spherical mirrors which can converge or diverge the parallel rays of light which fall on them. Please note that the spherical mirrors have a curve-like surface, so they are also known as curved mirrors. We will first define the spherical mirrors and then study the reflection of light from these spherical mirrors.
A spherical mirror is that mirror whose reflecting surface is the part of a hollow sphere of glass. The spherical mirrors are of two types : Concave mirrors, and Convex mirrors.
(i) A concave mirror is that spherical mirror in which the reflection of light takes place at the concave surface (or bent-in surface). A concave mirror is shown in Figure(a), in which the concave reflecting surface has been marked A. The other surface B in Figure (a), having short, oblique lines is the non-reflecting surface.
The inner shining surface of a steel spoon is an example of concave mirror (see Figure). In our ray-diagrams, we use only the side-view of a concave mirror as shown in Figure (a). If, however, we look at a concave mirror from the front, it appears to be like a piece of thin round glass whose front surface is shining and bent inward whereas back surface is covered with a paint and bulging outward.
(ii) A convex mirror is that spherical mirror in which the reflection of light takes place at the convex surface (or bulging-out surface). A convex mirror is shown in Figure (b), in which the convex reflecting surface has been marked B. The other surface A in Figure (b), having short, oblique lines is the non-reflecting surface.
The back-side of a shining steel spoon is an example of convex mirror (see Figure). In our ray diagrams, we use only the side-view of a convex mirror as shown in Figure (b). If, however, we look at a convex mirror from the front, it appears like a piece of thin, round glass whose front shining surface is bulging outward but the back surface covered with paint is bent inward.
A spherical mirror (concave mirror or convex mirror) works on the reflection of light. We will now understand the meaning of some new terms such as centre of curvature, radius of curvature, pole, and principal axis, which are used in the study of spherical mirrors.
Centre of Curvature, Radius of Curvature, Pole and Principal Axis of a Spherical Mirror
The centre of curvature of a spherical mirror is the centre of the hollow sphere of glass of which the mirror is a part. The centre of curvature of a mirror is represented by the letter C. In Figure (a), C is the centre of curvature of the concave mirror and in Figure (b), C is the centre of curvature of the convex mirror.
The centre of curvature is not a part of the mirror. It lies outside the reflecting surface of the mirror. It should be noted that the centre of curvature of a concave mirror is in front of it but the centre of curvature of a convex mirror is behind it.
The radius of curvature of a spherical mirror is the radius of the hollow sphere of glass of which the mirror is a part. In Figure (a), the distance CP is the radius of curvature of the concave mirror and in Figure (b), the distance CP is the radius of curvature of the convex mirror. The radius of curvature of a mirror is represented by the letter R.
The centre of a spherical mirror is called its pole. In other words, the middle point of a spherical mirror is called its pole. In Figure(a), P is the pole of the concave mirror and in Figure (b), P is the pole of the convex mirror. The pole of a spherical mirror lies on the surface of the mirror.
The straight line passing through the centre of curvature and pole of a spherical mirror is called its principal axis. In Figure (a), C is the centre of curvature of the concave mirror and P is the pole of the concave mirror, so the line XY, passing through C and P is the principal axis of the concave mirror. Similarly, in Figure (b), XY is the principal axis of the convex mirror. The principal axis is normal (or perpendicular) to the mirror at its pole.
That portion of a mirror from which the reflection of light actually takes place is called the aperture of the mirror. The aperture of a spherical mirror is represented by the diameter of its reflecting surface. In Figure, the distance MM’is the aperture of the mirror. In fact, the aperture of a mirror represents the size of the mirror.
Principal Focus and Focal Length of a Concave Mirror
The principal focus of a concave mirror is a point on its principal axis to which all the light rays which are parallel and close to the axis, converge after reflection from the concave mirror. Look at Figure in which a parallel beam of light rays is falling on a concave mirror MM’. In Figure, point F is the principal focus of the concave mirror because all the parallel rays of light converge at this point after getting reflected from the concave mirror.
Since all the reflected light rays actually pass through the focus (F) of a concave mirror, therefore, a concave mirror has a real focus. The focus of a concave mirror is in front of the mirror. Since a concave mirror converges a parallel beam of light rays, it is also called a converging mirror.
The focal length of a concave mirror is the distance between its pole and principal focus. In Figure, P is the pole of the concave mirror and F is the principal focus, so the distance PF is the focal length of this concave mirror. The focal length of a mirror is denoted by the letter.
We will now describe how an image of the sun can be formed by a concave mirror. The sun is at a far off distance from us, so the sunlight rays reaching us are parallel rays. An image of the sun can be produced by a concave mirror at its focus (see Figure).
This can be done by performing a simple experiment as follows. We take a concave mirror and point it towards the sun. Fold a piece of paper in front of the concave mirror in such a way that the sunlight reflected by concave mirror falls on the paper (see Figure). A small patch of bright reflected light will appear on the paper.
Adjust the distance of paper from the concave mirror in such a way that the sharpest point of bright light is obtained. This sharp point of light on paper is the image of sun formed by the concave mirror (see Figure). This image of the sun is formed at the focus of concave mirror (where the paper is held by us). If we keep this piece of paper in this position for a few minutes, the paper would start burning at the point of sun’s image and a hole will be formed in it.
This is because the concave mirror converges (or concentrates) a lot of sun’s rays to a small point on paper. The heat energy of these concentrated sun rays burns the paper. Please note that the image of the sun formed by the concave mirror is real because it can be received on screen (such as a sheet of paper).
Since the sun’s image is formed at the focus of the concave mirror, therefore, the distance of sun’s image (or paper) from the concave mirror gives us an approximate value of the focal length of the concave mirror.
Principal Focus and Focal Length of a Convex Mirror
The principal focus of a convex mirror is a point on its principal axis from which a beam of light rays, initially parallel to the axis, appears to diverge after being reflected from the convex mirror. In Figure, a parallel beam of light rays is incident on a convex mirror MM’. Each ray of light is reflected by the convex mirror, and the reflected rays diverge (spread out) from the mirror surface.
Let us produce all the reflected rays backwards (as shown by dotted lines) so that they appear to meet at a point F behind the convex mirror. Now, to a person looking into the mirror from the left side, all the reflected rays appear to be coming (or diverging) from the same point F behind the convex mirror.
This point F is the principal focus of the convex mirror. It should be noted that the reflected rays do not actually pass through the focus (F) of a convex mirror, therefore, a convex mirror has virtual focus.
Another point to be noted is that the focus of a convex mirror is situated behind the mirror. The focal length of a convex mirror is the distance from the pole P to its principal focus F. Thus, in Figure, the distance PF is the focal length of the convex mirror.
A plane mirror neither converges parallel rays of light nor diverges them. The focal length of a plane mirror can be considered to be ‘infinite’ or ‘infinity’ (which means very, very great or limitless).
Relation between Radius of Curvature and Focal Length of a Spherical Mirror
For a spherical mirror having small aperture, the principal focus (F) lies exactly mid-way between the pole (P) and centre of curvature (C) (see abobe Figures). So, the focal length of a spherical mirror (a concave mirror or a convex mirror) is equal to half of its radius of curvature. If f is the focal length of a spherical mirror and R is its radius of curvature, then :
f = \(\frac{R}{2}\)
Let us solve one problem now.
Example Problem.
If the radius of curvature of a spherical mirror is 20 cm, what is its focal length ?
Solution:
We know that : f = \(\frac{R}{2}\)
Here, Focal length, f = ? (To be calculated)
And, Radius of curvature, R = 20 cm
So, f = \(\frac{20}{2}\) cm
= 10 cm
Thus, the focal length of this spherical mirror is 10 cm.