SPHERICAL MIRRORS
A spherical mirror, whose reflecting surface is curved inwards, that is, faces towards the centre of the sphere, is called a `"concave mirror."`
A spherical mirror whose reflecting surface is curved outwards, is called a convex mirror. The schematic representation of these mirrors is shown in Fig. 10.1. You may note in these diagrams that the back of the mirror is shaded.
You may now understand that the surface of the spoon curved inwards can be approximated to a concave mirror and the surface of the spoon bulged outwards can be approximated to a `"convex mirror."`
Before we move further on spherical mirrors, we need to recognise and understand the meaning of a few terms. These terms are commonly used in discussions about spherical mirrors.
The centre of the reflecting surface of a spherical mirror is a point called the pole. It lies on the surface of the mirror. The pole is usually represented by the letter P.
The reflecting surface of a spherical mirror forms a part of a sphere. This sphere has a centre. This point is called the centre of curvature of the spherical mirror. It is represented by the letter C.
Please note that the centre of curvature is not a part of the mirror. It lies outside its reflecting surface.
The centre of curvature of a concave mirror lies in front of it. However, it lies behind the mirror in case of a convex mirror.
You may note this in Fig.10.2 (a) and (b). The radius of the sphere of which the reflecting surface of a spherical mirror forms a part, is called the radius of curvature of the mirror. It is represented by the letter R.
You may note that the distance PC is equal to the radius of curvature. Imagine a straight line passing through the pole and the centre of curvature of a spherical mirror. This line is called the principal axis.
`ul"Remember"` that principal axis is normal to the mirror at its pole. Let us understand an important term related to mirrors, through an Activity.
`ul"Activity 10.2"`
`"CAUTION :"` Do not look at the Sun directly or even into a mirror reflecting sunlight. It may damage your eyes.
♦ Hold a concave mirror in your hand and direct its reflecting surface towards the Sun.
♦ Direct the light reflected by the mirror on to a sheet of paper held close to the mirror.
♦ Move the sheet of paper back and forth gradually until you find on the paper sheet a bright, sharp spot of light.
♦ Hold the mirror and the paper in the same position for a few minutes. What do you observe? Why?
The paper at first begins to burn producing smoke. Eventually it may even catch fire. Why does it burn? The light from the Sun is converged at a point, as a sharp, bright spot by the mirror.
In fact, this spot of light is the image of the Sun on the sheet of paper. This point is the focus of the concave mirror. The heat produced due to the concentration of sunlight ignites the paper.
The distance of this image from the position of the mirror gives the approximate value of focal length of the mirror. Let us try to understand this observation with the help of a ray diagram.
Observe Fig.10.2 (a) closely. A number of rays parallel to the principal axis are falling on a concave mirror. Observe the reflected rays. They are all meeting/intersecting at a point on the principal axis of the mirror.
This point is called the principal focus of the concave mirror.
Similarly, observe Fig. 10.2 (b). How are the rays parallel to the principal axis, reflected by a convex mirror? The reflected rays appear to come from a point on the principal axis.
This point is called the principal focus of the convex mirror. The principal focus is represented by the letter F. The distance between the pole and the principal focus of a spherical mirror is called the focal length. It is represented by the letter f.
The reflecting surface of a spherical mirror is by and large spherical. The surface, then, has a circular outline. The diameter of the reflecting surface of spherical mirror is called its aperture. In Fig.10.2, distance `M N` represents the aperture.
We shall consider in our discussion only such spherical mirrors whose aperture is much smaller than its radius of curvature.
Is there a relationship between the radius of curvature `R`, and focal?
Is there a relationship between the radius of curvature `R`, and focal length `f`, of a spherical mirror?
For spherical mirrors of small apertures, the radius of curvature is found to be equal to twice the focal length. We put this as `R = 2f` . This implies that the principal focus of a spherical mirror lies midway between the pole and centre of curvature.
`ul" Image Formation by Spherical Mirrors"`
You have studied about the image formation by plane mirrors. You also know the nature, position and relative size of the images formed by them.
How about the images formed by spherical mirrors? How can we locate the image formed by a concave mirror for different positions of the object? Are the images real or virtual? Are they enlarged, diminished or have the same size? We shall explore this with an Activity.
`ul"Activity 10.3"`
You have already learnt a way of determining the focal length of a concave mirror. In Activity 10.2, you have seen that the sharp bright spot of light you got on the paper is, in fact, the image of the Sun.
It was a tiny, real, inverted image. You got the approximate focal length of the concave mirror by measuring the distance of the image from the mirror.
♦ Take a concave mirror. Find out its approximate focal length in the way described above. Note down the value of focal length. (You can also find it out by obtaining image of a distant object on a sheet of paper.)
♦ Mark a line on a Table with a chalk. Place the concave mirror on a stand. Place the stand over the line such that its pole lies over the line.
♦ Draw with a chalk two more lines parallel to the previous line such that the distance between any two successive lines is equal to the focal length of the mirror. These lines will now correspond to the positions of the points P, F and C, respectively.
`bb"Remember –"` For a spherical mirror of small aperture, the principal focus F lies mid-way between the pole P and the centre of curvature C.
♦ Keep a bright object, say a burning candle, at a position far beyond C. Place a paper screen and move it in front of the mirror till you obtain a sharp bright image of the candle flame on it.
♦ Observe the image carefully. Note down its nature, position and relative size with respect to the object size.
♦ Repeat the activity by placing the candle – (a) just beyond C, (b) at C, (c) between F and C, (d) at F, and (e) between P and F.
♦ In one of the cases, you may not get the image on the screen. Identify the position of the object in such a case. Then, look for its virtual image in the mirror itself.
♦ Note down and tabulate your observations.
You will see in the above Activity that the nature, position and size of the image formed by a concave mirror depends on the position of the object in relation to points P, F and C.
The image formed is real for some positions of the object. It is found to be a virtual image for a certain other position. The image is either magnified, reduced or has the same size, depending on the position of the object.
A summary of these observations is given for your reference in Table 10.1.
`ul"Representation of Images Formed by Spherical Mirrors Using Ray Diagrams"`
We can also study the formation of images by spherical mirrors by drawing ray diagrams. Consider an extended object, of finite size, placed in front of a spherical mirror. Each small portion of the extended object acts like a point source.
An infinite number of rays originate from each of these points. To construct the ray diagrams, in order to locate the image of an object, an arbitrarily large number of rays emanating from a point could be considered.
However, it is more convenient to consider only two rays, for the sake of clarity of the ray diagram. These rays are so chosen that it is easy to know their directions after reflection from the mirror.
The intersection of at least two reflected rays give the position of image of the point object. Any two of the following rays can be considered for locating the image.
(i) `"A ray parallel to the principal axis, after reflection,"` will pass through the principal focus in case of a concave mirror or appear to diverge from the principal focus in case of a convex mirror. This is illustrated in Fig.10.3 (a) and (b).
(ii) `"A ray passing through the principal focus of a concave mirror"` or a ray which is directed towards the principal focus of a convex mirror, after reflection, will emerge parallel to the principal axis. This is illustrated in Fig.10.4 (a) and (b).
(iii) `"A ray passing through the centre of curvature of a concave mirror"` or directed in the direction of the centre of curvature of a convex mirror, after reflection, is reflected back along the same path. This is illustrated in Fig.10.5 (a) and (b).
The light rays come back along the same path because the incident rays fall on the mirror along the normal to the reflecting surface.
(iv) `"A ray incident obliquely to the principal axis, towards a point P (pole of the mirror),"` on the concave mirror [Fig. 10.6 (a)] or a convex mirror [Fig. 10.6 (b)], is reflected obliquely.
The incident and reflected rays follow the laws of reflection at the point of incidence (point P), making equal angles with the principal axis.
`ul"Remember"` that in all the above cases the laws of reflection are followed. At the point of incidence, the incident ray is reflected in such a way that the angle of reflection equals the angle of incidence.
(a) `bb"Image formation by Concave Mirror"`
Figure 10.7 illustrates the ray diagrams for the formation of image by a concave mirror for various positions of the object.
`ul"Activity 10.4"`
♦ Draw neat ray diagrams for each position of the object shown in Table 10.1.
♦ You may take any two of the rays mentioned in the previous section for locating the image.
♦ Compare your diagram with those given in Fig. 10.7.
♦ Describe the nature, position and relative size of the image formed in each case.
♦ Tabulate the results in a convenient format.
`"Uses of concave mirrors"`
Concave mirrors are commonly used in torches, search-lights and vehicles headlights to get powerful parallel beams of light. They are often used as shaving mirrors to see a larger image of the face.
The dentists use concave mirrors to see large images of the teeth of patients. Large concave mirrors are used to concentrate sunlight to produce heat in solar furnaces.
`"(b) Image formation by a Convex Mirror"`
We studied the image formation by a concave mirror. Now we shall study the formation of image by a convex mirror.
`ul"Activity 10.5"`
♦ Take a convex mirror. Hold it in one hand.
♦ Hold a pencil in the upright position in the other hand.
♦ Observe the image of the pencil in the mirror. Is the image erect or inverted? Is it diminished or enlarged?
♦ Move the pencil away from the mirror slowly. Does the image become smaller or larger?
♦ Repeat this Activity carefully. State whether the image will move closer to or farther away from the focus as the object is moved away from the mirror?
We consider two positions of the object for studying the image formed by a convex mirror. First is when the object is at infinity and the second position is when the object is at a finite distance from the mirror.
The ray diagrams for the formation of image by a convex mirror for these two positions of the object are shown in Fig.10.8 (a) and (b), respectively. The results are summarised in Table 10.2.
You have so far studied the image formation by a plane mirror, a concave mirror and a convex mirror. Which of these mirrors will give the full image of a large object? Let us explore through an Activity.
`ul"Activity 10.6"`
♦ Observe the image of a distant object, say a distant tree, in a plane mirror.
♦ Could you see a full-length image?
♦ Try with plane mirrors of different sizes. Did you see the entire object in the image?
♦ Repeat this Activity with a concave mirror. Did the mirror show full length image of the object?
♦ Now try using a convex mirror. Did you succeed? Explain your observations with reason.
You can see a full-length image of a tall building/tree in a small convex mirror. One such mirror is fitted in a wall of Agra Fort.
If you visit the Agra Fort, try to observe the full-length image of a distant, tall building/tomb in the wall mirror. To view the tomb distinctly, you should stand suitably at the terrace adjoining the wall.
`bbul"Uses of convex mirrors"`
Convex mirrors are commonly used as rear-view (wing) mirrors in vehicles. These mirrors are fitted on the sides of the vehicle, enabling the driver to see traffic behind him/her to facilitate safe driving.
Convex mirrors are preferred because they always give an erect, though diminished, image. Also, they have a wider field of view as they are curved outwards.
Thus, convex mirrors enable the driver to view much larger area than would be possible with a plane mirror.
`ulbb"Sign Convention for Reflection by Spherical Mirrors"`
While dealing with the reflection of light by spherical mirrors, we shall follow a set of sign conventions called the New Cartesian Sign Convention. In this convention, the pole (P) of the mirror is taken as the origin.
The principal axis of the mirror is taken as the x-axis (X’X) of the coordinate system. The conventions are as follows –
(i) The object is always placed to the left of the mirror. This implies that the light from the object falls on the mirror from the left-hand side.
(ii) All distances parallel to the principal axis are measured from the pole of the mirror.
(iii) All the distances measured to the right of the origin (along + x-axis) are taken as positive while those measured to the left of the origin (along – x-axis) are taken as negative.
(iv) Distances measured perpendicular to and above the principal axis
(along + y-axis) are taken as positive.
(v) Distances measured perpendicular to and below the principal axis (along –y-axis) are taken as negative.
The New Cartesian Sign Convention described above is illustrated in Fig.10.9 for your reference. These sign conventions are applied to obtain the mirror formula and solve related numerical problems.
`ul"Mirror Formula and Magnification"`
In a spherical mirror, the distance of the object from its pole is called the object distance (u). The distance of the image from the pole of the mirror is called the image distance (v).
You already know that the distance of the principal focus from the pole is called the focal length (f). There is a relationship between these three quantities given by the mirror formula which is expressed as
` 1/v + 1/u = 1/f` .................(10.1)
This formula is valid in all situations for all spherical mirrors for all positions of the object. You must use the New Cartesian Sign Convention while substituting numerical values for u, v, f, and R in the mirror formula for solving problems.
`bbul"Magnification"`
Magnification produced by a spherical mirror gives the relative extent to which the image of an object is magnified with respect to the object size. It is expressed as the ratio of the height of the image to the height of the object.
It is usually represented by the letter m. If h is the height of the object and h′ is the height of the image, then the magnification m produced by a spherical mirror is given by
` m = text( Height of the image h' )/text( Height of the object h)`
` m = (h')/h \ \ \ \ .......(10.2)`
The magnification m is also related to the object distance (u) and image distance (v). It can be expressed as:
Magnification `(m) = (h')/h =- v/u \ \ \ \ .......(10.3)`
You may note that the height of the object is taken to be positive as the object is usually placed above the principal axis. The height of the image should be taken as positive for virtual images.
However, it is to be taken as negative for real images. A negative sign in the value of the magnification indicates that the image is real. A positive sign in the value of the magnification indicates that the image is virtual.
