Physics Topics such as mechanics, thermodynamics, and electromagnetism are fundamental to many other scientific fields.

## State the Rules for Obtaining Images Formed by Convex Mirror

In order to construct ray-diagrams to find out the position, nature and size of the images formed by a convex mirror, we should remember the paths of the following rays of light. We can call them the rules for obtaining images in convex mirrors.

Rule 1. A ray of light which is parallel to the principal axis of a convex mirror, appears to be coming from its focus after reflection from the mirror. This is shown in Figure. In Figure, the ray of light AB is parallel to the principal axis XP of a convex mirror.

The ray of light AB gets reflected at point B on the mirror and goes in the direction BD. To a person on the left side, the reflected ray BD appears to be coming from the focus F of the convex mirror situated behind the mirror (as shown by dotted line).

Rule 2. A ray of light going towards the centre of curvature of a convex mirror is reflected back along the same path. This is shown in Figure. In Figure, the ray of light AD is going towards the centre of curvature C of a convex mirror. It strikes the mirror surface at point D and gets reflected back along the same path DA.

To a person on the left side, the reflected ray DA appears to be coming from the centre of curvature C (as shown by dotted line behind the mirror). Please note that the ray AD gets reflected back along the same path because it falls normally (or perpendicularly) on the mirror surface at point D.

Please note that if a ray of light is incident on a convex mirror along its principal axis, then it is reflected back along the same path (because it will be normal or perpendicular to the mirror surface). The angle of incidence as well as the angle of reflection for such a ray of light will be zero.

Rule 3. A ray of light going towards the focus of a convex mirror becomes parallel to the principal axis after reflection. This is just the reverse case of the first rule and it is shown in Figure. Here the incident ray of light AE is going towards the focus F of the convex mirror. It strikes the mirror surface at point E and gets reflected. After reflection, it becomes parallel to the principal axis and goes in the direction EG (see Figure).

Rule 4. A ray of light which is incident at the pole of a convex 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 on the pole P of the convex mirror making an angle of incidence i with the principal axis XP. It gets reflected along the direction PH making an equal angle of reflection r with the principal axis (see Figure).

It should be noted that a convex mirror has its focus and centre of curvature behind it. Since no real rays -of light can go behind the convex mirror, all the rays shown behind the convex mirror are virtual (or unreal) and hence they have been represented by dotted lines. In fact, no actual rays can pass through the focus and centre of curvature of a convex mirror.

Please note that whatever be the position of object in front of a convex mirror, the image formed by a convex mirror is always behind the mirror, it is virtual, erect and smaller than the object (or diminished). When the distance of the object is changed from convex mirror, then only the position and size of the image changes. There are two main positions of an object in the case of a convex mirror from the point of view of position and size of image. The object can be :

- anywhere between pole (P) and infinity, and
- at infinity.

We will discuss both these cases one by one. Let us first describe the formation of image by a convex mirror when the object is placed between pole (P) of the mirror and infinity.

### Formation of Image by a Convex Mirror

In Figure, we have an object AB placed in front of a convex mirror M anywhere between pole P and infinity. A ray of light AD, parallel to the principal axis of the convex mirror, strikes the mirror at point D. Now, according to the first rule of image formation, this parallel ray of light should appear to be coming from focus f after reflection.

So, we join the points D and F by a dotted line and produce the line FD towards the left in the direction DX. Now, DX gives us the reflected ray which appears to be coming from focus F of the convex mirror.

We have now to draw a second ray of light from the point A going towards the centre of curvature C of the convex mirror. For this we join the point A with point C by a line which cuts the mirror at point E. The line from A to E is a solid line and it represents a real ray of light but the line from E to C is a dotted line which represents a virtual ray of light.

Now, AE represents a ray of light going towards the centre of curvature C of the convex mirror. According to the second rule of image formation, this ray is reflected along the same path EA but it appears to be coming from the centre of curvature C.

The two reflected rays DX and EA are diverging rays but they appear to intersect at point A’ when produced backwards. Thus A’ is the virtual image of point A of the object. To get the full image of the object, we draw the perpendicular A’B’ to the axis from point A’.

Thus A’B’ is the virtual image of the object AB. It is clear from Figure 49 that the image is formed behind the convex mirror between the pole and the focus. It is virtual, erect and smaller than the object (or diminished). From the above discussion we conclude that: When an object is placed anywhere between pole (P) and infinity in front of a convex mirror, the image formed is :

- behind the mirror between pole (P) and focus (F),
- virtual and erect, and
- diminished (smaller than the object).

If we hold a matchstick (as object) in front of a convex mirror, a virtual, erect and diminished (smaller) image of the matchstick is seen on looking into the convex mirror [see Figure(a)]. We know that the back side of a shining steel spoon (which is bulging outwards) is a kind of convex mirror.

So, if we keep our face in front of the back side of a shining steel spoon we will see a virtual and erect image of the face which is smaller in size as compared to the face [see Figure(b)].

If we move the object more and more away from the pole of the convex mirror, the image becomes smaller and smaller in size and moves away from the mirror towards its focus but it remains virtual and erect for all the positions of the object. And when the object is at infinity, the image is formed at the focus. This is discussed below.

### When the Object is at Infinity

When the object is at a far-off distance, we say that the object is at infinity. In Figure, we have a convex mirror M. Suppose an object (an arrow pointing upwards) has been placed at infinity in front of the convex 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.

The ray AD gets reflected in the direction DX and the ray AP gets reflected in the direction PY. When the diverging reflected rays DX and PY are produced backwards (as shown by dotted lines in Figure), they intersect at point A’ in the focal plane of the convex mirror.

Thus, A’ is the virtual image of the top point A of the object. To get the full image of the object, we draw A’B’ perpendicular to the axis. So, A’B’ is the image of the object AB placed at infinity. We find that the image is formed at the focus (F) of the convex mirror behind the mirror. It is virtual, erect and highly diminished (much smaller than the object). From the above discussion we conclude that: When an object is at infinity from a convex mirror, the image formed is :

- behind the mirror at focus (F),
- virtual and erect, and
- highly diminished (much smaller than the object).

Since the image of a distant object formed by a convex mirror is highly diminished, we can see the full image of a tall building or tree even in a small convex mirror. Please note that when the object kept at infinity in front of a convex 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 convex mirror is round in shape, then its image is formed at the focus according to the ray diagram shown in Figure on page 178. And before we conclude this discussion, here is a summary of the images formed by a convex mirror.

Summary of the Images Formed by a Convex Mirror

### Uses of Convex Mirror

(i) Convex mirrors are used as rear-view mirrors in vehicles (like cars, trucks and buses) to see the traffic at the rear side (or back side) (see Figure). A driver prefers to use a convex mirror as rear-view mirror because of two reasons :

(a) A convex mirror always produces an erect (right side up) image of the objects.

(b) The image formed in a convex mirror is highly diminished or much smaller than the object, due to which a convex mirror gives a wide field of view (of the traffic behind) [see Figure(a)]. A convex mirror enables a driver to view much larger area of the traffic behind him than would be Figure.

Convex mirror is used as a rear possible with a plane mirror. A plane mirror gives a narrow view mirror (or side view mirror) in vehicles, field of view [see Figure(b)]. Due to this if a plane mirror is used as a rear-view mirror in vehicles, it will give the driver a much smaller view of the road and traffic behind. Rear-view mirrors are also known as driving mirrors or side-view mirrors or wing mirrors.

Please note that we cannot use a concave mirror as a rear- view mirror in motor vehicles. This is because a concave mirror produces inverted images (upside down images) of distant objects. So, if we use a concave mirror as a rear view mirror in a car (bus or truck, etc.) we will see in the mirror that all the vehicles on the road (at the back side) are running upside down with their wheels up in the air. This will be really a very funny situation to watch.

(ii) Big convex mirrors are used as ‘shop security mirrors’ (see Figure). By installing a big convex mirror at a strategic point in the shop, the shop owner can keep an eye on the customers to look for thieves and shoplifters among them.

### How to Distinguish Between a Plane Mirror, a Concave Mirror and a Convex Mirror Without Touching Them

We can distinguish between these mirrors just by looking into them, that is, by bringing our face close to each mirror, turn by turn. All of them will produce an image of our face but of different types. A plane mirror will produce an image of the same size as our face and we will look our normal self.

A concave mirror will produce a magnified image and our face will look much bigger (like that of a giant !). A convex mirror will produce a diminished image and our face will look much smaller (like that of a small child !). Let us answer one question now.

**Example Problem.**

No matter how far you stand from a mirror, your image appears erect. The mirror may be :

(i) plane

(ii) concave

(iii) convex

(iv) either plane or convex

Choose the correct alternative.

**Answer:**

The correct alternative is :

(iv) either plane or convex.

### Numerical Problems Based On Convex Mirrors

In order to solve the numerical problems based on convex mirrors, we should remember that the mirror formula for a convex mirror is the same as that for a concave mirror, which is :

\(\frac{1}{v}\) + \(\frac{1}{u}\) = \(\frac{1}{f}\)

where v = image distance

u = object distance

and f = focal length

Please note that since a convex mirror always forms an image behind the mirror (on the right side of the mirror), therefore, the image distance (v) in the case of convex mirrors is always positive. The object is always placed to the left of mirror, so the object distance (u) is always taken with a negative sign.

The focus of a convex mirror is behind the mirror on its right side, therefore, the focal length of a convex mirror is always taken as positive.

The two magnification formulae for convex mirror are also just the same as that for a concave mirror. That is :

m = \(\frac{h_2}{h_1}\) and m = – \(\frac{v}{u}\)

where the symbols have their usual meaning. Let us now solve one numerical problem based on convex mirror.

**Example Problem.**

An object 5 cm high is placed at a distance of 10 cm from a convex mirror of radius of curvature 30 cm. Find the nature, position and size of the image.

**Solution:**

Here, Object distance, u = – 10 cm (To left of mirror)

Image distance, v = ? (To be calculated)

And, Focal length, f = \(\frac{\text { Radius of curvature }}{2}\)

= \(\frac{30}{2}\) cm

= + 15 cm (Convex mirror)

Putting these values in the mirror formula :

\(\frac{1}{v}\) + \(\frac{1}{u}\) = \(\frac{1}{f}\)

we get : \(\frac{1}{v}\) + \(\frac{1}{-10}\) = \(\frac{1}{+15}\)

or \(\frac{1}{v}\) – \(\frac{1}{10}\) = \(\frac{1}{15}\)

or \(\frac{1}{v}\) = \(\frac{1}{15}\) + \(\frac{1}{10}\)

\(\frac{1}{v}\) = \(\frac{2+3}{30}\)

\(\frac{1}{v}\) = \(\frac{5}{30}\)

\(\frac{1}{v}\) = \(\frac{1}{6}\)

So, Image distance, v = + 6 cm

Thus, the position of image is 6 cm behind the convex mirror. Since the image is formed behind the convex mirror, its nature will be virtual and erect.

To find the size of image, we will calculate the magnification first.

Now, Magnification, m = –\(\frac{v}{u}\)

Here, Image distance, v = + 6 cm

and Object distance, u = – 10 cm

So, m = – \(\frac{(+6)}{(-10)}\)

m = \(\frac{6}{10}\)

Magnification, m = 0.6

We also have another formula for magnification, which is :

m = \(\frac{h_2}{h_1}\)

Here, Magnification, m = + 0.6

Height of image, h_{2} = ? (To be calculated)

and Height of object, h_{1} = + 5 cm

So, +0.6 = \(\frac{h_2}{+5}\)

h_{2} = 5 × 0.6

Height of image, h_{2} = 3 cm (or + 3 cm)

Thus, the size of image is 3 cm.