# Mirrors. Equation of Curved Mirrors. Image Formation in Plane and Curved Mirrors

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12.8Mirrors. Equation of Curved Mirrors. Image Formation in Plane and Curved Mirrors

In this Physics tutorial, you will learn:

• What are mirrors?
• How many types of mirrors are there?
• How is the image formed in plane mirrors?
• Why do we use special rays to build the image in curved mirrors
• How is the image formed in curved mirrors?
• How to draw the image formed in mirrors?
• How many types of images are there?
• What are the characteristics of images produced in plane and curved mirrors?
• What is the mirrors equation?
• How to find the magnification produced by curved mirrors?
• What if there is a system composed by two or more mirrors?

## Introduction to Mirrors. Equation of Curved Mirrors. Image Formation in Plane and Curved Mirrors

What size and features has the image of your body if you look at yourself on a plane mirror? What happens to the image if you raise your left hand?

What kind of mirror must you use in order to see a wider space than usual? How is the image in this case?

Can you burn a paper using a mirror? What kind of mirror must it be?

In which direction in respect to each other do the incident rays coming from the Sun fall on a mirror? What happens with them after this?

These questions and many others will be explained during the following paragraphs of this tutorial, which deals with the fundamentals of Geometrical Optics - one of the most fascinating parts of Optics.

## Recap on Light Reflection

In the previous Optics tutorials, we have explained that:

• Light propagates in straight lines despite being a wave. This is because there is no single wave travelling in space but a bundle of waves instead, called ray when very thin, or beam when the thickness is considerable or when light enlarges in space.
• Mirrors are reflecting surfaces used to change the direction of light.
• When light falls on a plane mirror, it is reflected back, i.e. it passes again in air when hitting the smooth surface of a plane mirror. This reflection is regular, i.e. we are able to see not only the amount (intensity) of the reflected light but also the image of objects around. This cannot occur when light hits a rough surface because it is reflected in a diffuse way, i.e. only the intensity of light is still the same but it is not possible to obtain any image of the surrounding objects.
• The incident ray, the reflected ray and the normal line lie all at the same plane (First law of Reflection).
• The angle of incidence is equal to the angle of reflection (Second law of Reflection).
• Reflection of light occurs when a ray moving through the original medium (usually air) strikes a second medium (usually a smooth solid or liquid surface such as glass, still water etc.) and since it cannot penetrate through, it turns again to the original medium at the same pattern as shown in the figure below.

However, there are many other things to explain in this topic. First, it must be noted that the image formed in all plane mirrors has the same shape as the original object. Thus, a sphere will result in a spherical image in the mirror, a cylinder will give a cylindrical image, the reflection of a human body will result in the production of same kind of image on a mirror and so on. Therefore, if the shape of the original object is known, it is not necessary to use a very large number of rays to build the image after the reflection takes place but only a few rays emerging from the object's extremities. This saves us a lot of precious time during the study of objects' images on mirrors.

## Types of Mirrors

To explain mirrors and how different types of mirrors differ from each other, we will use a parallel incident beam coming from infinity. A light beam coming from the Sun would be a good approximation in this regard, as the Sun is far enough to be considered as being at infinity.

Given this, we can divide mirrors in two main categories:

### 1 - Plane mirrors

.

In Plane mirrors, an incident bundle of parallel rays coming from infinity will produce parallel rays even after reflection, as shown in the figure of the previous paragraph.

### 2 - Curved mirrors

.

Curved mirrors are formed by a part of a spherical shaped mirror. This category of mirrors contains two sub-categories in itself. They are:

1. Concave mirrors, in which only the inner part of the curved reflecting surface is reflective. In concave mirrors, a parallel beam coming from infinity will converge at a single point called focus, which is located at half-distance between the mirror and the geometrical centre of the sphere from which the mirror originates. When incident rays initially converg, they collect at a point, which is nearer than focus after reflection, and when incident rays are initially diverging, they collect at a point that is farther than focus after reflection.
2. Convex mirrors, in which only the outer part of the curved reflecting surface is reflective. In convex mirrors, a beam of parallel rays coming from infinity will diverge after striking the mirror in such a way that the extension of these rays passes through focus.

## Image Formation in Plane Mirrors

As stated earlier, we need to consider only rays coming out from the object's extremities to build up an image in mirrors. Given this, we obtain the following figure for the image formation in plane mirrors.

Thus, light rays that start from the extremities of object are reflected from the mirror and enter to the eye pupil as shown in the figure. The extension of the reflected rays give the object's image. Since this image is obtained by the extensions of reflected rays and not by the reflected rays themselves, it is called "virtual image". Its dimensions are equal to those of the original object and the distance from the mirror is the same as the distance from object to mirror (do = di).

As a result, we can see the image obtained through reflection by plane mirrors even though the object is not in our direct sight.

It must be noted that the image produced by plane mirrors is laterally inverted. This means when we are in front of a plane mirror and raise the left hand, the image will raise the right hand.

## Image Formation in Concave Mirrors

First, it is worth stating that all rays falling on a mirror whatever it may be (plane, concave or convex) are reflected at the same angle in respect to the normal line drawn from the mirror at the point in which the light is incident. Given this, we can identify four special rays among thousands of possible rays an object may produce. These special rays help us identify the position of image because the intersection of two of them gives us the position of the image's extremity.

More specifically, these four special rays are:

1. The ray originating from the higher extremity of object and which is incident to the mirror in parallel to its axis of symmetry, otherwise known as principal axis. After touching the mirror, it is reflected through focus as shown earlier.
2. The ray originating from the higher extremity of object, which touches the mirror at middle (at the origin of principal axis) and then is reflected at the same angle to the other side of symmetry axis.
3. The ray originating from the higher extremity of object, which first passes through focus and then is reflected in parallel to the principal axis after touching the mirror (the inverse of ray 1).
4. The ray originating from the higher extremity of object, which passes through the centre of curvature (twice the distance of focus). After touching the mirror it turns back because radius of sphere is normal to its inner surface at the point of the sphere in which it is incident.

The four rays discussed above are shown in the figure below.

Only two of the above special rays are enough to build up an image. There are six possible cases in image formation at concave mirrors based on the position of the object in respect to the mirror. They are:

1. The object is beyond the centre of curvature (do > 2F). The special rays 1 and 4 are enough to build the image as shown in the figure. You can easily see that the image is diminished (smaller) and vertically inverted in respect to the object. This image is real as it is obtained by the reflected rays, not by their extensions, just like in plane mirrors.
2. The object is at centre C of curvature, i.e. twice as far as the focus (do = 2F). We cannot use anymore the ray 4 but we can still use the other three special rays to build the image. Here let's use for example rays 1 and 3 for this purpose. The image in this case is equal in size as the object but it is vertically inverted. It is formed at the same position (at centre of curvature) and is real because it is obtained by the reflected rays, not by their extensions.
3. The object is located between the centre of curvature and focus (2F < do < F). We can use for example the rays 1 and 2 to build the image as shown below. The image produced in this case is larger than the object. It is formed beyond the centre of curvature (dî > 2F) and it is real because it is formed by the reflected rays, not by their extensions.
4. The object is located at focus of the concave mirror (do = F). In this case, the reflected rays are parallel to each other. This means no image is produced (in the language of mathematics we say 'the image is formed at infinity, as two parallel lines meet at infinity').
5. The object is located closer to the concave mirror than focus (do < F). We can use again the special rays 1 and 2 to build up the image. As you see, we cannot use the reflected rays to build up the image because they diverge from each other. Therefore, we use their extensions, which converge behind the mirror. As a result, an erect (upright) and larger image is formed at the position shown. Since this image is not obtained from the reflected rays but from their extensions, we say it is a virtual image.
6. The object is at infinity. In this case, only a parallel bundle of rays comes from the object to the mirror. As a result, the image will be simply a bright dimensionless point at focus, just like when we direct the mirror towards the sunlight.

## Image Formation in Convex Mirrors

Since in convex mirrors the centre of curvature and focus are on the other side of reflecting surface, there is only a single case of image formation, as there are no divisions on the object's placement side.

We use two rays to build up the image in convex mirrors. The first ray starts from the extremity of the object, moves parallel to the principal axis and is reflected in such a direction that its extension passes through focus. The other ray originates from the object's extremity, points to the centre of curvature but it turns back because it cannot penetrate through the mirror. However, we take its extension to build up the image, as the image is formed at the meeting point of the two extensions of the abovementioned rays as shown in the figure.

As you can see from the figure, the image is diminished, it is formed on the other side of the mirror, closer than focus and it is erect (upright). Since the image is not obtained from the reflected rays but from their extensions instead, it is a virtual image.

### Example 1

The image of an object is formed in a curved mirror as shown in the figure.

1. What kind of mirror is it?
2. Where is the object located approximately?

Take the object as placed on the principal axis.

### Solution 1

1. The mirror is concave, as in convex mirrors the image is always formed closer to the mirror than focus.
2. The situation shown in the figure represents the third case of image formation in concave mirrors as the image as the image is formed at a distance that is greater than the centre C of curvature. This means the image is enlarged and inverted. Therefore, the object is located between the centre of curvature and focus, in erect position as shown in the figure below.

## Equation of Curved Mirrors

There is an equation that provides the numerical relationship between the object's distance do, image distance and focal length (focus) F, which allows us to determine the position of image without having need for drawings. This equation (known as the Equation of Curved Mirrors) is

1/do + 1/dî = 1/F

However, this is not that simple; not always all values are positive. Thus, we need to correctly apply the sign rules if we want to avoid mistakes in the position values. These rules are provided below.

1. The object's position do is always taken as positive.
2. In concave mirrors, the focal length F is taken as positive while in convex mirrors it is taken as negative.
3. If the image is real, its position is taken as positive, while if the image is virtual its distance is taken as negative.

### Example 2

An object is placed 12 cm in front of a concave mirror of focal length equal to 18 cm.

1. What are the image features for this case?
2. What is the position of image formed in this mirror?

### Solution 2

1. From the given values, it is clear that the image is formed nearer to the concave mirror than focus. Therefore, this situation represents the fifth case of image formation in concave mirrors. It means the image is virtual, erect and enlarged.
2. We have the following clues based on the description and the sign rules of image formation in curved mirrors.
do = 12 cm
f = 18 cm
d1 = ?

Applying the mirror equation

1/do + 1/dî = 1/F

we obtain after substitutions

1/12 + 1/dî= 1/18
1/dî= 1/18 - 1/12
= 2/36 - 3/36
= - 1/36

Therefore, the position of image is dî = -36 cm. This means it is formed at 36 cm behind the mirror, as shown in the figure below.

## Magnification of Curved Mirrors

In daily life, magnification M is calculated by dividing the height of the image to the height of the original object. In symbols, we have:

M = himage/hobject

However, applying the triangle similarity rules, we can use another formula for the magnification of curved mirrors. It considers the image and object's position and does not require any information about the height of the object. We have

M = di/do

If we consider the example in the previous paragraph, we obtain for the magnification of the concave mirror:

M = -36 cm/12 cm
= -3

The above result means the image's height is triple the height of the original object. The sign minus indicates that the image is virtual.

## What If There Are More Than Two Curved Mirrors in the Same System?

In this case, the calculations are performed by considering the mirrors one by one. This means the image produced by the first mirror acts like an object for the second mirror and so on. The rules are the same as for a single curved mirror.

### Example 3

: An object is placed between two mirrors - one concave and the other convex - which are 1.5 m apart from each other. The object is 40 cm in front of the concave mirror. If both mirrors have the same focal length equal to 30 cm, calculate the position of the second image produced on the convex mirror (i.e. the image produced on the convex mirror when the image on the concave mirror is considered as object for the convex mirror).

### Solution 3

First, let's determine the position of the image produced in the concave mirror in respect to it. We have the following clues:

do = 40 cm
F1 = 30 cm
dî = ?

We have:

1/do +1/dî = 1/F
1/40 + 1/dî = 1/30
1/dî = 1/30 - 1/40
= 4/120 - 3/120
= 1/120

Therefore, the image produced in the concave mirror is 120 cm away from it. This means this image - which acts as an object for the convex mirror - is 150 cm - 120 cm = 30 cm before the convex mirror.

Given that in convex mirrors the focus is negative, we have the following clues for the second reflection:

do = 30 cm
F2 = - 30 cm
dî = ?

We have:

1/do +1/dî = 1/F
1/30 + 1/dî= - 1/30
1/dî= - 1/30 - 1/30
= - 2/30
= - 1/15

Note: The fraction - 2/30 is simplified to - 1/15. Thus, the second image is produced 15 cm after the convex mirror. Look at the figure.

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