# Mirror Formula

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### Mirror Formula (Concave Mirror)

Mirrors are a part of our life! It is quite natural for all of us to look at ourselves when we pass across a mirror. As common as it is to find a mirror or to look at our image, the science behind the mirror is actually fascinating. The mirrors which we use in our daily life are the ‘flat’ or the ‘plane’ mirrors (they are not curved), and any object in front of it appears exactly the same size in the mirror. The distance of the object’s image is also formed exactly at the same distance behind the mirror as the distance the object is in front of the mirror. This phenomenon is explained by the laws of reflection. Laws of reflection applies to both flat mirrors and curved mirrors.
The reflection of an object in a curved mirror forms an image which is not necessarily located at the same distance as the object’s distance. Also, the image size is not necessarily the same as the object’s size. However, these details can be estimated by using the Mirror Formula. In order to understand the mirror formula, let us first take a look at how an image is formed in a concave mirror.
Curved Mirrors:
The basic curved mirrors can be considered as a part of the spherical mirrors. It appears as though a spherical mirror has been sliced thus forming the two basic types of curved mirrors. When the inner curved surface is silvered to form a reflecting surface then it is the concave mirror, and when the outer surface of the curve is silvered to form a reflecting surface, then it is the convex mirror.
The two basic types of curved mirrors are:
i.            Concave Mirror: A mirror that is curved inward and has a center that goes inward.
ii.            Convex Mirror: A mirror that is curved outward, and has a center that comes outward.

### What is a Concave Mirror?

A Concave mirror is a mirror that is curved inward, and hence has the center that goes inward. A simple way to remember this is by taking the word ‘cave’ from ‘concave’, and think that concave mirrors cave inward, just like you are looking inside a cave! For a concave mirror, the inner surface of the curve is silvered so that it forms the reflecting surface. In order to understand reflection upon a concave mirror, there are important terms to be understood.

Center of Curvature: Considering that a concave mirror is actually a part of a sphere, the point which is the center of this sphere is known as the Center of Curvature ‘C’.
Principal Axis: The line connecting the center of the sphere and the center of the mirror is the known as the Principal Axis.
Vertex (or Pole): The point where the Principal Axis meets the mirror (the reflecting surface) is known as the Vertex or the Pole of the mirror. The vertex can also be defined as the center of the mirror taken geometrically. In the figure on the right, it is represented by the letter ‘P’.
Radius of curvature: The distance from the center of curvature to the vertex is known as the radius of curvature, R.
Focal point: The midpoint between the center of curvature and the vertex is known as the Focal point ‘F’.
Focal distance (or focal length): The distance from the mirror to the focal point is known as the Focal distance, ‘f’.

### The Law of Reflection:

Our eyes make it possible to see everything around us. But then why in dark, any object even in front of our eyes is not visible? This is because of the absence of light. When light falls on an object, it is reflected back and this reflected light rays reach our eyes and hence make the object visible to us. The Law of Reflection explains this behavior of light, and this law can be applied to both plane mirrors and curved mirrors.

·         The ray of light that approaches a mirror is known as the Incident ray. The point at which the light ray strikes the mirror is known as the Point of Incidence.
·         The ray of light that returns back from the mirror is known as the Reflected ray.
·         At the point of incidence, the line drawn perpendicular to the surface of the mirror is known as the Normal line.
·         The angle formed in between the incident ray and the reflected ray is divided by this normal line, and therefore 2 angles are formed.
·         The angle formed in between the incident ray and the normal is known as the Angle of Incidence, and the angle formed in between the normal and the reflected ray is known as the Angle of Reflection.

The Law of Reflection states that when a light ray reflects off a surface (flat or curved), then the angle of incidence is equal to the angle of reflection
Angle of Incidence = Angle of Reflection

Reflection in a Concave Mirror:
The law of reflection of light is applied to both flat mirrors and curved mirrors. The reflection law helps us to determine the location of the image for a particular object. This image location is observed as the point where all the reflected light rays appear to be diverging. It is not an easy task to use the law of reflection in a curved mirror and estimate the image location. In order to do this, the normal line which is the line drawn at the point of incidence perpendicular to the curved surface must be drawn and then the law of reflection must be applied to determine the image location.
The 2 rules which help us find the image location in a concave mirror are:
i.            If any incident ray travels parallel to the principal axis, then after reflection on the concave mirror its reflected ray will pass through the focal point.
ii.            If any incident ray passes through the focal point, then after reflection on the concave mirror its reflected ray will travel parallel to the principal axis.

### Characteristics of Image formed in Concave Mirrors:

In order to determine the size of the image, orientation, location and the type of image formed by the reflection of the object in the concave mirrors, ray diagrams were constructed. Using the ray diagrams, the characteristics of the image for a certain object location can be described.
Some of the important characteristics that are commonly analyzed from ray diagrams are:
·         Location of the image relative to the object’s location.
·         Orientation of the image – whether the image is upright or is inverted.
·         Size of the image relative to the object’s size – whether it is reduced, magnified or is the same size as the object.
·         Type of image formed – whether the image is a real image or is a virtual image.
There are 5 general locations where the object is placed, and with respect to these locations the characteristics of the image are observed.
Location 1: The object is placed beyond the Center of Curvature

·         If the object is placed beyond the center of curvature ‘C’, then the image will be formed at a location in between the center of curvature and the focal point.
·         The orientation of the image is inverted, and therefore in this case an inverted image is formed.
·         The size of the image is reduced, which implies that the image formed is smaller in dimensions than the object’s dimensions.
·         The type of image formed in this case is a real image. This is because the light rays converge at the image location due to which a real image is formed. To observe this practically, a sheet of paper can be placed at the location of the image and it will be observed that the replica of the object (reduced in size) would appear on the paper.

Location 2: The object is placed at the center of curvature

·         If the object is located at the center of curvature ‘C’, then the image is also formed at the center of curvature.
·         The orientation of the image is inverted, and therefore in this case an inverted image is formed.
·         The size of the image is exactly the same as the size of the object. Therefore the object and image have same dimensions.
·         The image formed in this case is a real image as the light rays converge at the location of the image. This implies that the formed image can be practically observed on the sheet of paper when the paper is placed at the image location.

Location 3: The object is located in between the center of curvature and the focal point

·         If the object is located in anywhere between the center of curvature ‘C’ and the focal point ‘F’, then the image is formed beyond the center of curvature.
·         The orientation of the image is inverted and therefore in this case an inverted image is formed.
·         The size of the image is magnified in this case. This implies that the formed image will have dimensions greater than the object’s dimensions.
·         The image formed in this case is a real image. This is because the rays of light converge at the location of the image and therefore the image can be practically observed on a sheet of paper when the paper is placed at the image location.

Location 4: The object is placed at the focal point

·         If the object is located at the focal point, then there is no image formed. This is because the rays of light from the focal point (which also happen to be the object’s location) will reflect upon the mirror and will neither converge nor diverge. After reflection, these reflected rays travel parallel to each other and therefore no image is formed.

Location 5: The object is placed in front of the focal point

·         If the object is placed anywhere in front of the focal point, then the image will be always formed on the opposite side of the concave mirror.
·         The orientation of the image is upright, which implies that the image is not inverted and is upright just like the object.
·         The size of the image formed in this case is magnified. This implies that the dimensions of the image are greater than the dimensions of the object.
·         The type of image formed in this case is a virtual image. This is because the rays of light after reflection upon the concave mirror diverge. In order to get the point of intersection of these diverging rays, the reflected rays are extended backwards and this takes us behind the mirror. This point of intersection of the reflected rays is the image location and since it is formed behind the mirror, hence it is a virtual image. In this case there will be no image formed on the sheet of paper as light does not actually pass through the location of the image.
Mirror Formula:
Ray diagrams are extremely helpful in trying to understand and to determine the location of the image, its orientation, size, and the type of image formed.  However, the ray diagrams do not give us the numerical details such as how far the image is located from the object etc. In order to find the numerical details we use the Mirror Formula.
Mirror Formula: 1/f = 1/do + 1/di
Here f = focal length
do = object distance
di = image distance
Sign Convention:
·         f = positive, if the mirror is a concave mirror (it is negative if it is a convex mirror).
·         di = positive, if it is a real image located on the same side as the object.
·         di = negative, if it is a virtual image and is located behind the mirror.
Magnification formula:
The ratio of the height of the image and the height of the object is equal to the ratio of the image distance and the object distance. This equation is known as the Magnification equation.
Magnification, M = hi/ ho = - di/do
Here, hi = image height
ho = object height
di = image distance
do = object distance
Sign Convention:
·         hi = positive, if the image is upright (this also implies that the image in the concave mirror is virtual)
·         hi = negative, if the image is inverted (this also implies that the image in the concave mirror is real)

Example: A 3.00cm tall lamp is placed at a distance of 26.4cm from the concave mirror. The concave mirror has a focal length of 15.00cm. Determine the image distance and the size of the image.

Given information: ho = 3.00cm
do = 26.4cm
f = 15cm
If the focal length of the concave mirror is 15cm, then it implies that the center of curvature which is double the focal length is at 30cm.
From the given object distance and focal length, we can observe that the object is actually placed in between the center of curvature and the focal point.
Applying the Mirror Formula we have: 1/f = 1/do + 1/di
1/15 = 1/26.4 + 1/di
This implies: 1/di = 1/15 – 1/26.4
Taking the common denominator we get: 1/di = (26.4 – 15)/ 396
1/di = 11.4/396
So, di = 396/26.4 ==> di = 34.7cm (approximately)
This implies that the image distance is 34.7cm
Now, to find the image size we use the Magnification equation
Magnification, M = hi/ ho = - di/do
This implies: hi/ 3.00 = - 34.7/ 26.4
hi = - 3.94cm (approximately)
Therefore the size of the image is - 3.94cm (negative value implies that the image is inverted).
Magnification, M = hi/ ho
Hence, M = 3.94/ 3 = 1.31 (nearly)
Now, from the calculated image distance, di we can observe that the location of the image is far beyond the center of curvature and this fits in perfectly well with our above mentioned “Location 3 - the object located in between the center of curvature and the focal point”.
Also, we can observe that the image formed is magnified and since it is inverted it is also a real image!