Mirrors. Equation of Curved Mirrors. Image Formation in Plane and Curved Mirrors Revision Notes

In these revision notes for Mirrors. Equation of Curved Mirrors. Image Formation in Plane and Curved Mirrors, we cover the following key points:

• 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?

Mirrors. Equation of Curved Mirrors. Image Formation in Plane and Curved Mirrors Revision Notes

Mirrors are reflecting surfaces used to change the direction of light.

We can divide mirrors in two main categories:

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

2 a) 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.

2 b) 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.

The image produced by plane mirrors is laterally inverted. 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 (d_o = d_i).

There are four special rays used to build the image in spherical mirrors. They 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. 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 (d_o > 2F). In this case, the image is diminished (is 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 (d_o = 2F). 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 < d_o < F). 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.
5. The object is located at focus of the concave mirror (d_o = 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').
6. The object is located closer to the concave mirror than focus (d_o < F). Since the reflected rays diverge from each other, 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.
7. 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.

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. The image formed by convex mirrors 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.

There is an equation that provides the numerical relationship between the object's distance d_o, image distance dî 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

We must apply the sign rules to avoid mistakes in calculations. These rules are:

1. The object's position d_o 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 dî is taken as positive, while if the image is virtual its distance dî is taken as negative.

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:

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. Thus, we have

If there are two more curved mirrors in the same system, 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.