Mirror formula

Introduction

A mirror formula is a way to predict the relationship between three critical points of the mirror – the distance of the object, the distance of the image and the mirror’s focal length. The distance of the object placed is represented by “u”, the distance of the image formed is denoted by “v” and the focal length of the mirror is represented by “f”. The mirror formula can be applied to 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 and the mirror’s focal length, the mirror formula will help us 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 the mirror formula in detail.

What Is the Mirror Equation?

The mirror formula relates three aspects to each other – the object distance, the focal length of the mirror and the image distance. Here is the definition of these three aspects:

• The distance of the object from the reflecting mirror surface is known as the object’s distance. This distance is denoted by “u”.
• The distance of the image from the reflecting mirror surface is known as the distance of the image. This distance is denoted by “v”. 
• The distance between the principal focus and pole of the mirror is defined as the focal length of the mirror. It is denoted by “f”. In addition to this, the mirror’s focal length is half of the radius of curvature of the spherical mirror. The radius of curvature is denoted by the letter “r”. Here is the relation between these two

 f = r/2.

Thus, the mirror formula is given by: 1/v + 1/u = 1/f  

or 1/v + 1/u = 2/r.  

Sign Conventions for Mirror Equation 

Here are the sign conventions for the mirror equation. To measure the different distances, the optical centre of the lens is considered the centre point.

• The distances in the same direction of the incident light are considered to have a positive sign. 
• The distances in the opposite direction of the incident light are considered to have a negative sign. 
• The heights in an upward and perpendicular direction to the principal axis are considered positive. 
• In contrast, the heights that are downward and at the same time perpendicular to the principal axis are known to have negative signs. 
• Along the x-axis, the principal axis of the mirror is placed. The pole of the mirror is considered to be the origin.
• At the left-hand side of the mirror, the object is placed. From this side, the incident light is stroked. 
• The distance parallel to the principal axis is measured from the pole of the mirror. 

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

Mirror Equation for a Concave Mirror

The mirror equation for a 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.  

Assumptions for Derivation of Mirror Formula 

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

• All the distances are measured from the pole of the mirror. 
• As per the sign conventions, the opposite of the distance to the incident beam of light is considered to have a negative sign, whereas the beam of light in the same direction is known to have a positive sign. 
• 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 the Mirror Equation

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

Example – 

The radius of curvature of the convex mirror used in the rear view mirror of a vehicle is 4.00 m. A bus is located at 6.00 m from the mirror. Find the formed image’s position.

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

Substituting the values 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 will be formed 1.5 m behind the mirror. 

Applications of Mirror Equation

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

• When the distance of the object and focal length of the mirror are known, the distance of the image can be easily predicted. 
• When the distance of the image and focal length of the mirror are known to us, the distance of the object can be easily found using the mirror formula. 
• The mirror formula or the mirror equation can easily find the mirror’s focal length. The condition is that both the distance of the image and that of the object should be accurately known to us. 
• When the mirror equation or mirror formula is used along with the magnification equation, the height of the object or height of the image can be obtained.

• Conclusion

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