Contents
- 1 Different Types of Image Formation in Concave Mirror
- 1.1 Formation of Different Types of Images By A Concave Mirror
- 1.2 Case 1. Image formed by a concave mirror when the object is placed between pole and focus of the mirror (Object between P and F)
- 1.3 Case 2. When the object is placed at the focus of a concave mirror (Object at F)
- 1.4 Case 3. When the object is placed between focus and centre of curvature (Object between F and C)
- 1.5 Case 4. When the object is placed at the centre of curvature of a concave mirror (Object at C)
- 1.6 Case 5. When the object is beyond the centre of curvature of the concave mirror (Object beyond C)
- 1.7 Case 6. When the Object is at Infinity
- 1.8 How to find out the Focal Length of a Concave Mirror Quickly but Approximately
- 1.9 Uses of Concave Mirrors
By learning Physics Topics, we can gain a deeper appreciation for the natural world and our place in it.
Different Types of Image Formation in Concave Mirror
When an object is placed in front of a concave mirror, an image is formed. The image is formed at that point where at least two reflected rays intersect (or appear to intersect).
Now, to find out the position of an image formed by a concave mirror, we will use only those two rays of light (starting from the object), whose paths, after reflection from the mirror, are known to us and easy to draw. The following rays of light are usually used to locate the images formed by concave mirrors. We can call them rules for obtaining images in concave mirrors.
Rule 1. A ray of light which is parallel to the principal axis of a concave mirror, passes through its focus after reflection from the mirror. This is shown in Figure. Here, we have a concave mirror M and its principal axis is XP. The centre of curvature is C and its focus is F.
Now, a ray of light AB is parallel to the principal axis of the mirror. It strikes the mirror at point B and gets reflected. After reflection its path changes and it passes through the focus F and goes in the direction BY (see Figure).
Rule 2. A ray of light passing through the centre of curvature of a concave mirror is reflected back along the same path (because it strikes the mirror normally or perpendicularly). This is shown in Figure.
A ray of light AD is passing through the centre of curvature C of the concave mirror. It falls normally (or perpendicularly) on the mirror at point D and gets reflected back along the same path DA. It should be noted that we have put two arrow-heads on the line AD which point in the opposite directions.
The arrow pointing from left to right indicates the direction of incident ray and the arrow pointing from right to left indicates the direction of reflected ray (see Figure). The ray of light passing through the centre of curvature of a concave mirror is reflected back along the same path because it strikes the concave mirror at right angles (90°) to its surface due to which the angle of incidence and angle of reflection both are 0°.
Ride 3. A ray of light passing through the focus of a concave mirror becomes parallel to the principal axis after reflection. This rule is just the reverse case of the first rule and it is shown in Figure. Here, the incident ray AE is passing through the focus F of the concave mirror. It strikes the mirror surface at point E and gets reflected. After reflection, it becomes parallel to the axis and goes in the direction EG.
Rule 4. A ray of light which is incident at the pole of a concave mirror is reflected back making the same angle with the principal axis. This is shown in Figure. Here a ray of light AP is incident at the pole P of the concave mirror making an angle of incidence i with the principal axis. It gets reflected along the direction PH such that the angle of reflection (r) is equal to the angle of incidence (i).
Please note that if a ray of light is incident on a concave mirror along its principal axis, then it is reflected back along the same path (because it will be normal to the mirror surface). The angle of incidence as well as the angle of reflection for such a ray of light will be zero.
Formation of Different Types of Images By A Concave Mirror
The type of image formed by a concave mirror depends on the position of object in front of the mirror. We can place the object at different positions (or different distances) from a concave mirror to get different types of images. For example, we can place the object :
- between the pole (P) and focus (F) (see Figure),
- at the focus (F),
- between focus (F) and centre of curvature (C),
- at the centre of curvature (C),
- beyond the centre of curvature (C), and
- at far-off distance called infinity (This distance cannot be shown curvature) in the Figure).
Please note that in all our ray-diagrams the object, an arrow AB pointing upwards, will be placed on the principal axis. So, a ray of light coming from the bottom B of the object always travels along the principal axis and gets reflected back along the same path. We, however, do not put arrows on the principal axis to show this ray of light.
We will now draw the ray-diagrams to show the images formed by a concave mirror for the different positions of an object. We will consider all the six positions of the object, one by one.
Case 1. Image formed by a concave mirror when the object is placed between pole and focus of the mirror (Object between P and F)
In Figure, we have an object AB placed between the pole (P) and focus (F) of a concave mirror, that is, the object is within the focus of the concave mirror. To find out the position and nature of the image, starting from A, we draw a ray AD parallel to the axis (see Figure). This ray gets reflected at D and then passes through the focus F. A second ray of light AE passing through the centre of curvature C strikes the mirror normally (or perpendicularly) at point E and gets reflected back along the same path.
Now, the two reflected rays DF and EC are diverging rays and, therefore, do not intersect each other on the left side. The reflected rays DF and EC are produced backwards (as shown by dotted lines). On producing backwards, they appear to intersect at point A’ behind the concave mirror.
Thus, A’ is the virtual image of point A of the object. To get the complete image of the object we draw A’B’ perpendicular to the axis from point A’. Thus A’B’ is the image of the object AB formed by the concave mirror. Now, an eye at position X, sees the reflected rays as if they have come from points A’B’ behind the mirror. Since there is no actual intersection of the reflected rays, so the image A’B’ is a virtual image.
This image can be seen only by looking into the concave mirror, it cannot be obtained on a screen. We can see from Figure that the image is same side up as object (both of them have arrow-head at the top), so it is an erect image. And if we compare the size of the object AB and its image A’B’, the image appears to be bigger in size than the object. Thus, it is an enlarged image or magnified image.
From the above discussion we conclude that: When an object is placed between the pole (P) and focus (F) of a concave mirror, the image formed is :
- behind the mirror,
- virtual and erect, and
- larger than the object (or magnified).
It is clear from the above discussion that to obtain a magnified and erect image with a concave mirror, the object should be placed between the pole and focus of the concave mirror, that is, the object should He placed within the focus (or focal length) of the concave mirror.
For example, if the focal length of a concave mirror is 10 cm, then its focus (F) will be at a distance of 10 cm from the pole of the mirror. So, to obtain a magnified and erect image with this concave mirror, we will have to place the object at a distance of less than 10 cm so that it remains between the pole and focus.
Thus, if we place the object at a distance of, say 8 cm, from this concave mirror, the object will be within the focus and a magnified, virtual and erect image will be formed behind the concave mirror (which can be seen by looking into the concave mirror).
If we hold a matchstick (as object) within the focus of a concave mirror, then a magnified, virtual and erect image of the matchstick is seen on looking into the concave mirror [as shown in Figure (a)],
This also explains the use of a concave mirror as a shaving mirror. When the face is placed close to a concave mirror (so that the face is within its focus) the concave mirror produces a magnified (large) and erect image of the face. Since a large image of the face is seen in the concave mirror, it becomes easier to make a smooth shave.
Thus, while using a concave mirror as a shaving mirror, the face should be close to the concave mirror. For the same reason, concave mirrors are also used as make-up mirrors by women for putting on make-up (such as painting eye-lashes) [see Figure (b)]. In order to use a concave mirror as a make-up mirror, the face is held close to the concave mirror so that it lies within the focus of concave mirror.
Dentists use concave mirrors to see the large image of the teeth for examining the teeth of a person (see Figure (c)], For this purpose, the dentist holds a small concave mirror in such a way that the tooth lies within its focus. A magnified image of the tooth is then seen by the dentist in the concave mirror. Since the tooth looks much bigger, it becomes easy to examine the defect in the tooth.
Case 2. When the object is placed at the focus of a concave mirror (Object at F)
In Figure, the object AB has been placed at the focus (F) of the concave mirror. Now, the parallel ray of light AD (coming from the top of the object) gets reflected at D and passes through the focus F, giving us the reflected ray DX.
A second ray of light AE passing through the centre of curvature C, is reflected back along the same path giving us another reflected ray EY. We find that the reflected rays DX and EY are parallel to one another.
These parallel rays will intersect (or meet) at a far off distance to form an image ‘at infinity’. And since the image is formed at infinity, it is not possible to show it in our diagram. From this discussion we conclude that: When an object is placed at the focus of a concave mirror, the image formed is :
- at infinity,
- real and inverted, and
- highly magnified (or highly enlarged).
Suppose we have a concave mirror of focal length 10 cm, then its focus (F) will be at a distance of 10 cm from it. So, by saying that an object is placed at the focus of this concave mirror, we will mean that the object is placed at a distance of 10 cm from the concave mirror. In this case, the concave mirror converts the diverging rays of light coming from the object into a parallel beam of light rays (which form image at infinity).
When a lighted bulb is placed at the focus of a concave mirror reflector, the diverging light rays of the bulb are collected by the concave reflector and then reflected to produce a strong, parallel-sided beam of light [see Figure (a)].
This explains the use of concave reflectors in torches, car head-lights, and search lights to produce a strong beam of light (which travels a considerable distance in the darkness of night).
Case 3. When the object is placed between focus and centre of curvature (Object between F and C)
In Figure, the object AB has been placed between the focus F and centre of curvature C of a concave mirror. Now, a ray of light AD parallel to the principal axis gets reflected at point D and then passes through the focus F. A second ray of light AE passing through the centre of curvature C falls normally on the mirror surface at E and returns along the same path.
Thus, we have two reflected rays DF and EC which are converging in the downward direction. If we extend these rays further in the downward direction, they actually intersect at point A’. Thus A’ is the real image of point A of the object. To get the complete image we draw A’B’ perpendicular to the axis from point A’.
Thus, A’B’ is the real image of the object AB and it can be received on a screen. For example, if we take a lighted candle as the object then the image of its wick will be formed on a white screen placed at the image position. If we look at the image A’B’, we find that it is wrong side up having arrow-head at its bottom. So, we say that the image is inverted. The size of image is larger than the object, so the image is magnified, and it has been formed beyond the centre of curvature of the mirror.
From the above discussion we conclude that: When an object is placed between the focus (F) and centre of curvature (C) of a concave mirror, the image formed is :
- beyond the centre of curvature,
- real and inverted, and
- larger than the object (or magnified).
Please remember that a real image is always inverted, and a virtual image is always erect. Suppose we have a concave mirror of focal length 10 cm. Then its focus (F) will be at a distance of 10 cm and centre of curvature (C) will be at a distance of 2 × 10 = 20 cm from it.
Now, by saying that the object is placed between the focus and centre of curvature of this concave mirror, we mean that the object is at any distance between 10 cm and 20 cm from the concave mirror.
For example, it may be at a distance of say, 15 cm from the concave mirror. And by saying that the image is formed beyond centre of curvature, we mean that it is at a distance greater than 20 cm from this concave mirror.
Case 4. When the object is placed at the centre of curvature of a concave mirror (Object at C)
In Figure, the object AB has been placed at the centre of curvature C of the concave mirror. A ray of light AD which is parallel to the principal axis passes through the focus F after reflection. Now, the second ray of light that we usually use is the one passing through the centre of curvature C.
But in this case the object itself is placed at the centre of curvature, so we cannot use this ray of light to locate the image. Here we will use rule No. 3 of image formation which says that “A ray of light passing through the focus of a concave mirror becomes parallel to the principal axis after reflection”.
So, we now take the ray AE passing through the focus F. It strikes the mirror at point E and gets reflected in the direction EA’ parallel to the principal axis.
The reflected rays DA’ and EA’ meet at point A’, so A’ is the real image of point A of the object. To get the complete image, we draw A’B’ perpendicular to the principal axis. Thus, A’B’ is the real image of the object AB (Note that B and B’ is just the same point).
The image is of the same size as the object, it is real, inverted and formed at the centre of curvature, where the object itself is placed. From this discussion we conclude that : When an object is placed at the centre of curvature (C) of a concave mirror, the image formed is :
- at the centre of curvature (C),
- real and inverted, and
- same size as the object.
Suppose we have a concave mirror of focal length 10 cm, then its focus (F) will be at a distance of 10 cm and its centre of curvature (C) will be at a distance of 2 × 10 = 20 cm.
So, by saying that the object is at the centre of curvature, we mean that the object is at a distance of 20 cm from this concave mirror. And by saying that the image is formed at the centre of curvature, we mean that the image is also formed at a distance of 20 cm from the concave mirror.
Case 5. When the object is beyond the centre of curvature of the concave mirror (Object beyond C)
In Figure 39, the object AB has been placed beyond the centre of curvature C of the concave mirror. A ray of light AD which is parallel to the principal axis, passes through the focus F after reflection. A second ray of light AE passing through the centre of curvature falls normally on the mirror surface at E and returns along the same path.
The two reflected rays intersect at A’. Thus, A’ is the real image of point A of the object. To get the complete image, we draw A’B’ perpendicular to the axis. Thus, A’B’ is the complete image of the object AB. It is clear from Figure that the image A’B’ is formed between the focus (F) and centre of curvature (C).
The image is real and inverted. It is smaller in size than the object AB, so we call it a diminished image. From the above discussion we conclude that: When an object is placed beyond the centre of curvature (C) of a concave mirror, the image formed is :
- between the focus and centre of curvature,
- real and inverted, and
- smaller than the object (or diminished).
Suppose we have a concave mirror of focal length 10 cm, then its focus (F) will be at a distance of 10 cm and its centre of curvature (C) will be at a distance of 2 × 10 = 20 cm from it. Now, by saying that an object is placed beyond the centre of curvature of this concave mirror, we mean that the object is placed at a distance greater than 20 cm (say, 25 cm) from the concave mirror. And by saying that the image is formed between focus and centre of cur vat Lire, we mean that the image is formed at a distance between 10 cm and 20 cm.
We know that the front side of a shining steel spoon is a kind of concave mirror. So, if we keep our face at a fairly good distance from the front side of a shining steel spoon (so that it is beyond its centre of curvature), we will see a real, inverted and smaller image (diminished image) of our face in the spoon (as shown in Figure)
Case 6. When the Object is at Infinity
When the object is at a very large distance, we say that the object is at infinity. In Figure we have a concave mirror M. Suppose an object (an arrow pointing upwards) has been placed at infinity in front of the concave mirror (Since the object is very far off, it cannot be shown in the diagram).
Because the object AB is very far off, the two rays AD and AP coming from its top point A are parallel to one-another but at an angle to the principal axis as shown in Figure.
These parallel rays get reflected at points D and P and then intersect at point A’ in the focal plane of the mirror. Thus, A’ is the real image of the top point A of the object. To get the full image of the object, we draw A’B’ perpendicular to the principal axis from A’. Thus, A’B’ is the image of the object AB placed at infinity.
We find that image A’B’ is formed at the focus (F) of the concave mirror. It is real, inverted and much smaller than the object or highly diminished. From the above discussion we conclude that : When an object is at infinity from a concave mirror, the image formed is :
- at the focus (F),
- real and inverted, and
- much smaller than the object (or highly diminished).
It should be noted that this case is just opposite of case No. 2 which we have already discussed. There we studied that “when the object is at focus of concave mirror, the image is formed at infinity” and here we have seen that “when the object is at infinity, the image is formed at the focus of concave mirror”. This means that a concave mirror can concentrate all the parallel rays of light to its focus.
Suppose we have a concave mirror of focal length 10 cm, then its focus (F) will be at a distance of 10 cm from it. Now, by saying that the object is at infinity, we mean that the object is at a distance which is many, many times its focal length : it may be 1000 cm, 10,000 cm or even more. And by saying that the image is formed at the focus of this concave mirror, we mean that the image is formed at a distance of 10 cm from the pole of this concave mirror.
Please note that when the object kept at infinity in front of a concave mirror is assumed to be a big arrow pointing upwards, then its image is formed at focus according to the ray-diagram shown in Figure. If, however, the object kept at infinity in front of a concave mirror is round in shape (like the sun), then its image is formed at the focus according to the ray-diagram shown in Figure on page 177.
The fact that when a parallel beam of light rays falls on a concave mirror, the concave mirror concentrates all the light at its focus, explains the use of concave mirror as a doctor’s head-mirror. A concave mirror is used as a “head-mirror” by the doctors to concentrate light coming from a lamp on to the body part of a patient (like ear, nose, throat, etc.) to be examined.
This is done as follows : A concave mirror is fixed to the front part of a doctor’s head with the help of a strap [as shown in Figure (a)], When the beam of light coming from a lamp falls on the doctor’s concave head-mirror, the mirror focusses this light on to the body part of the patient. The body part gets illuminated brightly with this light due to which the doctor can look into the body part (like ear, nose, throat, etc.) very clearly.
The concave ‘metal dishes’ are used in dish antennas (or dish aerials) of televisions to receive TV signals from the very distant communications satellites which are high up in the sky [see Figure(b)]. The dish is a concave reflector. The dish collects a large amount of parallel beams of TV signals coming from the far off satellite and converges them to its focus.
The antenna (or signal detector) is fixed in front of the concave dish at its focus. Since the antenna is fixed at the focus of dish, it receives the strongest possible TV signals from the satellite which make our television work. The TV signals coming from satellite can be made even more stronger by using a bigger dish in the dish antenna. Please note that though the TV signals coming from the satellite are not light rays, they obey the laws of reflection of light.
How to find out the Focal Length of a Concave Mirror Quickly but Approximately
The fact that “when the object is at a considerable distance (or at infinity) from a concave mirror, then its image is formed at the focus” can be used to find out the focal length of a concave mirror quickly but approximately. We focus a distant object (several metres away) like a window or tree on a screen by using a concave mirror whose focal length is to be determined (see Figure).
The sharp image of window or tree will be formed at the focus of the concave mirror. That is, the distance of image (or screen) from the concave mirror will be equal to the focal length of concave mirror. This distance can be measured with a scale. It will give us the approximate focal length of the concave mirror. And before we conclude this discussion, here is a summary of the images formed by a concave mirror.
Summary of the Images Formed by a Concave Mirror
Uses of Concave Mirrors
(i) Concave mirrors are used as shaving mirrors to see a large image of the face. This is because when the face is held within the focus of a concave mirror, then an enlarged image of the face is seen in the concave mirror. This helps in making a smooth shave. For the same reason, concave mirrors are used as make-up mirrors.
(ii) Concave mirrors are used by dentists to see the large images of the teeth of patients. This is because when a tooth is within the focus of a concave mirror, then an enlarged image of the tooth is seen in the concave mirror. Due to this, it becomes easier to locate the defect in the tooth. The concave mirrors used by dentists are very small in size. They are fitted in a frame with a long handle.
(iii) Concave mirrors are used as reflectors in torches, vehicle head-lights and search lights to get powerful beams of light. This is because when a lighted bulb is placed at the focus of a concave reflector, then the concave reflector produces a powerful beam of parallel light rays. This helps us see things up to a considerable distance in the darkness of night. Concave reflectors are also used in room heaters. The concave reflectors of room heaters direct heat rays into the whole room.
(iv) Concave mirrors are used as doctor’s head-mirrors to focus light coming from a lamp on to the body parts of a patient (such as eye, ear, nose, throat, etc.) to be examined by the doctor.
(v) Concave dishes are used in TV dish antennas to receive TV signals from the distant communications satellites. The concave dish collects a lot of TV signals and focusses them on to an antenna (or aerial) fixed at its focus, so as to produce strong signals to run the television.
(vi) Large concave mirrors are used in the field of solar energy to focus sun’s rays for heating solar furnaces. This can be explained as follows : The solar furnace is placed at the focus of a large concave reflector. The concave reflector focuses the sun’s heat rays on the furnace due to which the solar furnace gets very hot. Even steel can be melted in this solar furnace.
Let us solve some problems now.
Example Problem 1.
We wish to obtain an erect image of an object using a concave mirror of focal length 15 cm. What should be the range of distance of the object from the mirror ? What is the nature of the image ? Is the image larger or smaller than the object ? Draw a ray-diagram to show the image formation in this case.
Solution:
In order to obtain an erect image of an object with a concave mirror, the object should be at a distance less than its focal length. Here the focal length of concave mirror is 15 cm. So to obtain an erect image of the object by using this concave mirror, the object should be placed at any distance which is less than 15 cm from the mirror. The nature of image will be virtual. The image will be larger than the object (For ray-diagram, see Figure on page 181).
Example Problem 2.
The image formed by a concave mirror is seen to be virtual, erect and larger than the object. The position of the object must then be :
(i) between the focus and centre of curvature.
(ii) at the centre of curvature.
(iii) beyond the centre of curvature.
(iv) between the pole of the mirror and its focus.
Choose the correct alternative.
Solution:
The correct alternative is (iv) : between the pole of the mirror and its focus.
Example Problem 3.
A concave mirror has a focal length of 10 cm. Where should an object be placed in front of this concave mirror so as to obtain an image which is real, inverted and same size as the object ?
Solution:
When the image formed by a concave mirror is real, inverted and of the same size as the object, then the object must be placed at its centre of curvature (C). Now, the centre of curvature of a concave mirror is at a distance of ‘twice the focal length’ or ‘2f’. Here,
Focal length, f = 10 cm
So, 2f = 2 × 10 cm
= 20 cm
Thus, the object should be placed at a distance of 20 cm in front of this concave mirror.
Example Problem 4.
An object is placed at the following distances from a concave mirror of focal length 10 cm :
(a) 8 cm
(b) 15 cm
(c) 20 cm
(d) 25 cm
Which position of the object will produce :
(i) a diminished real image ?
(ii) a magnified real image ?
(iii) a magnified virtual image ?
(iv) an image of the same size as the object ?
Solution:
In this case the focal length of concave mirror is 10 cm. This means that the focus (F) of this concave mirror is at a distance of 10 cm from the mirror and its centre of curvature (C) is at a distance of 2 × 10 = 20 cm from the mirror.
(i) A diminished real image is formed by a concave mirror when the object is beyond C. Here C is at 20 cm. So, the diminished real image will be formed when the object is at a distance greater than 20 cm, which in this problem is 25 cm. So, the position of object for a diminished real image is 25 cm.
(ii) A magnified real image is formed by a concave mirror when the object is between F and C. Here F is at 10 cm and C is at 20 cm. So, the magnified real image will be formed when the object is at a distance between 10 cm and 20 cm, which in this problem is 15 cm. So, the position of object for a magnified real image is 15 cm.
(iii) A magnified virtual image is formed by a concave mirror when the object is within focus (F) at a distance less than focal length or less than 10 cm, which in this problem is 8 cm. So, the position of object for a magnified virtual image is 8 cm.
(iv) An image of the same size as object is formed by a concave mirror when the object is at centre of curvature (C). Here C is at 20 cm. So, the image of same size as the object will be formed when the object is at 20 cm from the concave mirror. Thus, the position of object for an image of same size as the object is 20 cm.
Please note that we have given so many details in answering the above question just to make you understand clearly. There is, however, no need to give so many details in the examination. In the examination, the answer to the above question can just be written as :
(i) 25 cm
(ii) 15 cm
(iii) 8 cm
(iv) 20 cm