Mirror Formula

Introduction  

A mirror formula is a way to predict the relationship between three important points of the mirror – that is – the distance of the object, the distance of image, and the focal length of the mirror. The distance of the object placed is represented by “u”, the distance of the image formed is represented by “v”, and the focal length of the mirror is represented by “f”. This mirror formula can be applied both on plane mirrors as well as spherical mirrors (both concave and convex mirrors). According to the mirror equation, if we know the placement of the object as well as the focal length, the mirror formula will help us to predict the place where the image will be formed. This image is real and is formed in front of the mirror. Let’s discuss everything related to mirror formulas in detail. 

Introduction To Mirror Formula 

The mirror formula is a formula that relates three things of the mirror to each other – object distance, the focal length of the mirror, and image distance. Here is the definition of the three points. 

1. The distance of the object from the reflecting mirror surface is known as the distance of the object. This distance is denoted by “u”. 
2. The distance of the image from the reflecting mirror surface is known as the distance of the image. This distance is denoted by “v”. 
3. The distance between the principal focus and the pole of the mirror is defined as the focal length of the mirror. It is denoted by “f”. In addition to this, the focal length of the mirror is half of the radius of the curvature. The radius of the curvature is denoted by the letter “r”. Here is the relation between the two:- f = r/2.

Thus, the mirror formula is given as follows :- 1/v + 1/u = 1/f 

 or 1/v + 1/u = 2/r. 

Sign Conventions For Mirror Formula 

Here are the sign conventions for the mirror Formula. 

1. To measure the different distances, the optical center lens is taken into consideration. 
2. The distances that are in the same direction of the incident light, are considered to have a positive sign. 
3. The distances that are in the opposite direction of the incident light, are considered to have a negative sign. 
4. The heights that are in an upward and perpendicular direction to the principal axis are considered to be positive. 
5. In contrast, the heights that are downward and at the same time perpendicular to the principal axis, are known to have negative signs. 
6. Along the x-axis, the principal axis of the mirror is placed. The pole of the mirror is considered to be the origin.
7. At the left-hand side of the mirror, the object is placed. From this side, the incident light is struck. 
8. The distances that are parallel to the principal axis are all measured from the pole of the mirror. 

Thus, the focal length is negative for concave mirrors, and the focal length is positive for convex mirrors. 

Mirror Equation for a concave mirror

The mirror equation for concave mirror and convex mirror is similar to each other and is represented as follows :

1/f = 1/u + 1/v 

Where f is the focal length of the mirror, u is the distance of the object and v represents the distance of the image. This equation is commonly known as the mirror equation that relates the three factors to each other. 

Assumptions For Derivation Of Mirror Formula 

When the aforementioned mirror formula was derived, there were certain assumptions that were made. Here are the assumptions for mirror formula derivation. 

1. All the distances are measured from the pole of the mirror. 
2. As per the sign conventions, the distances opposite to the incident beam of light are considered to have negative signs whereas the beam of light that is in the same direction is known to have a positive sign. 
3. The distance above the principal axis is considered to have positive signs. In contrast, the distance below the principal axis has negative signs. 

These were the three assumptions that were focused on during the derivation of the mirror formula. 

Example of Mirror Equation

Let’s understand the mirror formula with an example of a mirror equation. 

Example – 

Suppose the radius of the curvature of the convex mirror that is used in the rearview mirror of the vehicle is 4.00 m. There is a bus that is located 6.00 meters from the mirror. Find the image position that is formed. 

Solution – 

Given 

Radius of curvature, r = + 4.00 m

Distance of the image, v =? 

Distance of the object, u = – 6.00 m 

As per the mirror formula, 

f = r/2 = +4 / 2 = + 2 m

Substitute the values back in the formula, 

1/f = 1/u + 1/v

½ = ⅙ + 1/v 

On solving, v = 12/8 = 1.5 m

Thus, the image of the bus is 1.5 m behind the mirror. 

Applications of Mirror Equation

Here are a few applications where the mirror formula or mirror equation is used. 

1. When the distance of the object and the focal length of the mirror are both known, the distance of the image can be easily predicted. 
2. Similarly, when the distance of the image and the focal length of the mirror are both known to us, the distance of the object can be easily found from the mirror formula. 
3. Mirror formula or the mirror equation can easily find the focal length of the mirror. The condition is that both the distance of the image and that of the object should be accurately known to us. 
4. When the mirror equation or mirror formula is used along with the magnification equation, either the height of the object or the height of the image can be obtained. 

Conclusion 

A polished surface that is known to reflect the light that is incident on the reflecting surface is known as a mirror. The reflected and incident light will have multiple similar properties, such as wavelength and other physical properties. We hope the article has helped you to understand the mirror formula in a better way. We have also covered the sign conventions of the mirror formula along with the other main characteristics.