Mirror Equation in Physics is the equation for mirrors that provides the relation between the distance of the object and the image, as well as its focal length. Mirror Equation is helpful in determining object position, image position or focal length given that two of the parameters are given. In optics, which is the branch of science that deals with the study of light and its interactions with various materials and optical elements, mirrors are used in various instruments for various purposes. Thus, understanding the relationship between the focal length of the mirror and the distance between the object and the image from the mirror is very important.

In this article, we will explore the concept of the Mirror Equation in detail, Mirror Equation Proof, and Mirror Equation for Magnification with various types of mirrors as well. So, let’s start learning about the concept of Mirror Equations.

What is Mirror in Optics?

In optics, a mirror is an optical device which reflects the light and allows us to see objects by redirecting the light that strikes it. Mirrors have a smooth, highly reflective surface, typically made of glass with a thin metallic coating on the backside. The most common type of mirror is the flat or plane mirror, but there are also curved mirrors with different shapes and properties, such as concave and convex mirrors, which we will discuss further in the article.

What is Mirror Equation?

Mirror Equation is a formula that provides the relationship between the focal length, image distance, and object distance of any spherical mirror. The Mirror Equation is based on the principles of reflection and is given by

1/f = 1/v + 1/u

Where,

• f is the focal length of the mirror,
• v is the image distance i.e., distance of the image from the mirror, and
• u is the object distance i.e., distance of the object from the mirror.

Derive Mirror Equation

We can learn the proof of this formula in the article: Derivation of Mirror Formula

Mirror Magnification Equation

The magnification formula for mirrors relates the size and orientation of an object to the size and orientation of its image formed by a mirror. For mirrors magnification formula is given as:

M = – v/u

Where,

• M is the magnification,
• v is the image distance (distance of the image from the mirror), and
• u is the object distance (distance of the object from the mirror).

Sign Convention for Mirror Equation

A new Cartesian symbol scheme is used to avoid confusion in the understanding of ray directions. Look at the picture for a better view.

• The pole of the mirror is taken into account to measure all distances.
• Measuring distances left to the pole ‘P’ of mirror is considered negative.
• Distances are considered positive if they are right hand direction to the pole P.
• When measured upward and perpendicular to the principal axis, they are considered positive.
• If measured downward and perpendicular to the principal axis, they are considered negative.

Read more about Sign Convention for Spherical Mirrors.

Types of Mirrors

There are generally three types of mirrors in optics: 

• Plane Mirrors 
• Concave Mirrors
• Convex Mirrors

Let’s discuss these types in detail as follows:

Plane Mirror

A plane mirror is a type of mirror with a flat, reflective surface. Unlike concave mirrors, which have a curved inward reflective surface, plane mirrors have a completely flat reflective side. Plane mirrors do not have a focal point, and they do not converge or diverge incident rays of light.

Features of Plane Mirror

• The reflective surface is flat.
• Plane mirrors do not have a focal point.
• They produce virtual images that are the same size as the object and appear to be located behind the mirror. Plane Mirror Produce lateral inverted images.
• Commonly used in everyday items like bathroom mirrors, dressing mirrors, and rearview mirrors in vehicles.

Concave Mirror

A concave mirror is a mirror with a curved inward reflective surface, meaning the reflective side is facing inward. The centre of the mirror’s curvature is known as the focal point (F). Concave mirrors are converging mirrors, which means they can focus parallel incident rays of light onto a single point, called the focal point.

Features of Concave Mirror

• The reflective surface curves inward.
• The focal point is located on the same side as the mirror’s reflective surface.
• It can form real or virtual images, depending on the object’s position relative to the focal point.
• Used in applications such as reflecting telescopes, makeup mirrors, and headlights in vehicles.

Convex Mirror

A convex mirror is a mirror with a curved outward reflective surface, meaning the reflective side is facing outward. The centre of curvature is also the focal point, but in convex mirrors, the focal point is on the opposite side of the reflective surface.

Features of Convex Mirror

• Reflective surface curves outward.
• The focal point is located on the opposite side of the mirror’s reflective surface.
• It always forms virtual images, which are reduced in size compared to the actual object.
• Used in applications like security mirrors in stores and parking lots, side-view mirrors on vehicles, and some outdoor traffic mirrors.

Read More,

• Spherical Mirrors
• Types of Spherical Mirrors

Applications Of Mirror Equation

The mirror equation is used in various practical applications in real life. Here are some applications of the mirror equation:

• Image Formation: The mirror equation helps determine the characteristics of images formed by concave and convex mirrors. It predicts whether the image is real or virtual, upright or inverted, and magnified or reduced in size.
• Telescope Design: In reflecting telescopes that use concave mirrors as the primary focusing element, the mirror equation is crucial for designing and optimizing the telescope’s optical performance
• Camera Lenses: For cameras that utilize concave or convex mirrors or lenses, the mirror equation is used in designing the lens system and predicting the image’s properties.
• Headlights: The bulb of a headlight or a torch is placed at the focus of a concave mirror. The light rays, coming from the bulb, emerge out as parallel rays after getting reflected at the concave mirror. The reflected rays can cover large distances with high intensity.
• Optical cavity: Concave mirrors are used in optical cavities, which play an important role in laser physics. The light rays reflect multiple times inside a cavity and form standing waves. 
• Rear-view: Due to its reflective properties, it is used as a rear-view mirror in vehicles.

Mirror Equation Related Articles

• Convex and Concave Mirrors
• Reflection of Light
• Image Formation by Spherical Mirrors

Solved Examples on Mirror Equation

1. Find out the position of an object from a convex mirror of the focal length of 6 cm, which produces an image on the mirror axis at 3 cm from the mirror.

Solution:

Given data , f = 6 cm (convex mirror)

v = -3 cm

Using mirror equation, we have

1/u = 1/f-1/v

⇒ 1/u = ⅙ – (-⅓) 

⇒ 1/u = ⅙ + ⅓ = (1+2)/6 = 1/2

⇒ u = 2 cm

Thus, the position of the object is 2 cm from the mirror. 

2. Calculate the focal length of a concave mirror with a curvature radius of 25 cm.  

Solution:

Given that, R = 25 cm (Radius of curvature)

The radius of curvature of a concave mirror is twice its focal length. 

R = 2f

Where R is the radius of curvature of the mirror and F denotes focal length. 

So f = R/2 

Thus, f = 25/2

⇒ f = – 12.5

The negative sign here denotes it’s a concave mirror. 

3. An object is placed at a distance of 2 times of focal length from the pole of the convex mirror, calculate the linear magnification.

Solution:

Let the Focal length of mirror = f

So, the object distance, u = -2f 

The formula to calculate image distance we use mirror formula as,

1 / v + 1 / u = 1 / f

Therefore,

1 / v + 1 / -2f = 1 / f 

⇒ 1 / v  = 1 / f + 1 / 2f  

⇒ 1 / v  = 3 / 2f

⇒ v = 2f / 3

Magnification is given as, 

m = – v / u

⇒ m = -(2f/3) / (-2f) 

⇒ m =  1/3

4. If the image is a distance of 6 cm and the object is at 12 cm in the front of the concave mirror, calculate the magnification formed.

Solution:

Given that,

The distance of object, u = – 12 cm

The distance of image, v = – 6 cm

Since,

Magnification is given by, 

m = – v / u

⇒ m = – (-6 / -12) 

⇒ m = -0.5

Hence, the image will be diminished by nearly half as size of object. 
 

5. Calculate the image’s position if the bus is 5 m away from a convex mirror. The Convex Mirror has a 8 m radius of curvature. 

Solution:

Given that,

The curvature radius (R) is +8.00 m.

Distance between objects (u) = -5.00 m

We need to figure out what image distance(v) equals.

We know that f = R/2 = 8/2 = 4 m 

The mirror’s formula is 1/u + 1/v = 1/f

⇒ 1/v = 1/f – 1/u

⇒ 1/v = 1/4 – 1/(-5)

⇒ 1/v = 9/20

Thus, v = 20/9

2.22 meters ()

As a result, the picture is created 2.22 meters away from the mirror.

Practice Problems on Mirror Equation

Problem 1: An object is placed 15 cm in front of a concave mirror with a focal length of 10 cm. Determine the image distance and state whether the image is real or virtual.

Problem 2: A convex mirror has a focal length of -20 cm. If an object is placed 30 cm in front of the mirror, find the image distance and describe the nature of the image.

Problem 3: A concave mirror has a radius of curvature of 30 cm. Calculate the focal length of the mirror and determine the image distance when an object is placed 20 cm in front of it.

Problem 4: An object is positioned 12 cm in front of a concave mirror with a focal length of 8 cm. Calculate the magnification of the image.

Problem 5: A concave mirror has a radius of curvature of 40 cm. If an object is placed at a distance of 60 cm from the mirror, determine the image distance and whether the image is magnified or diminished.

Problem 6: A convex mirror has a focal length of 30 cm. If an object is positioned 40 cm in front of the mirror, calculate the image distance and describe the characteristics of the image.

Mirror Equation – FAQs

1. What are Mirrors?

Mirrors are reflective surfaces that bounce back light, allowing us to see our own reflection and objects in front of them. They are commonly made of glass with a reflective coating.

2. What are the 3 Types of Mirrors?

The three main types of mirrors are:

• Plane Mirrors
• Concave Mirrors
• Convex Mirrors

3. What are Concave and Convex Mirrors?

• Concave Mirrors: Curved inward, they can focus light or create magnified or diminished images.
• Convex Mirrors: Curved outward, they diverge light and create smaller, wider images.

4. What is the Mirror Equation?

The formula for mirror equation is 1/f = 1/v+1/u.

5. How does the Mirror Equation Work?

The mirror equation quantifies the relationship between the object distance (v), the image distance (u), and the focal length (f).

6. What are the Types of Images Produced by Spherical Mirrors?

The concave mirror produces both real and virtual images.

7. What is a Real Image?

A real image is a point where actual light rays converge after reflecting off a mirror or passing through a lens. It can be captured on a screen or observed directly.

8. What is a Virtual Image?

A virtual image is an apparent image that appears to be in a location where light rays appear to diverge or do not actually converge. It cannot be captured on a screen.

9. What is the Equation for Magnification of a Mirror?

The equation for magnification (M) of a mirror is:

M = – (image height) / (object height) = – (image distance)/(object distance)

In this equation, a negative sign indicates an inverted image, while a positive sign denotes an upright image.

10. What is Lens and Mirror Equation?

Lens Formula is given as 1/f = 1/v – 1/u where as Mirror Equation is Given as 1/f = 1/v + 1/u