Mirror Formula - What is Mirror Equation?

What is a Mirror?

The law of reflection states that when a ray of light is made to fall on the reflecting surface, the incident ray, the reflected ray, and the normal to the surface of the mirror all lie in the same plane and the angle of incidence is equal to the angle of reflection.

Types of Mirrors

The following are the most common types of mirrors:

Plane mirror

The reflected images in their normal proportions but reversed from left to right are formed by a plane mirror. 

Convex mirror

These are curved outward spherical mirrors, and the image obtained is virtual, diminished, and erect for a real object.

Concave mirrors

These are curved inward spherical mirrors, and the image obtained from these mirrors is dependent on the placement of the object.

What is Mirror Formula?

The mirror formula is the relationship between the focal length of the mirror, the object's distance u from the pole of the mirror, and the image's distance v from the pole.

The mirror equation is

1/v +1/u = 1/f.

Mirror Formula Explanation

The Mirror formula represents the relationship between the object distance (u), image distance (v), and focal length (f) of a Spherical Mirror. The mirror equation is written as follows:

1/f = 1/v+1/u

Here, u and v represent the distances of the object and image from the mirror's pole, respectively. And Focal Length (f) is the distance between the primary focus and the pole.

Using the equation below, we can calculate the magnification (m) of the object using u and v.

m=−v/u

The radius of curvature (R) is twice its focal length.

R=2f

and f=R/2 

Hence, the Mirror equation can be written as:

 1/u+1/v = 1/f= 2/R

Mirror Formula: New Sign Conventions

The new Cartesian Sign Convention is used to avoid confusion in understanding the ray directions. Please take a look at the diagram for a better understanding.

• The optical center of the lens is taken into account when measuring all distances.
• Distances are considered negative when measured in the opposite direction of the incident light.
• Distances are considered positive when measured in the same direction as the incident light.
• Height is considered positive when measured upwards and perpendicular to the principal axis.
• Height is considered negative when measured downwards and perpendicular to the principal axis.

Read More About: Mirror Formula Derivation

Applications of Mirror Equation

The Mirror Equation is used in the following ways:

• When the object distance and the focal length of the mirror are known, the mirror equation can be used to predict the image distance.
• When we know the image distance and the focal length of the mirror, we can use the equation to calculate the object distance.
• The mirror equation allows us to calculate the focal length of the mirror simply by knowing the distance between the object and the image it forms.
• When we use the mirror equation in conjunction with the magnification equation, we can obtain the value of either the image height or the object height when the other is given.

Things to Remember

• Optics is the branch of physics that deals with light, its behavior patterns, and qualities. 
• Light is a type of electromagnetic radiation that enables the human eye to see or renders objects visible. It is also known as radiation that can be seen with the naked eye.
• The focal length of a lens determines its ability to converge or diverge light rays.
• We can use mirror formulas and equations to determine where the image will be produced if we know the object position and the focal length of the mirror.
• A mathematical formula that connects the object distance, image distance, and mirror focal length is known as the Mirror Formula.
• The mirror formula is 1/f = 1/v + 1/u​

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Solved Examples of Mirror Formula

Q1. Calculate the magnification of a concave mirror if the image of an object formed in front of a concave mirror with a focal length of 12 cm is formed at a point 10 cm away from the mirror.:

Solution:

To calculate the magnification (M) of a concave mirror, we can use the mirror formula and the magnification formula. The mirror formula is:     
1/f=1/d_o + 1/d_i     
where:     
- f is the focal length of the mirror.     
- d_o is the object distance (the distance from the object to the mirror).     
- d_i is the image distance (the distance from the image to the mirror).     
The magnification of (M) is given by:     
M= h_i/h_o = -d_i/d_o     
where:     
- h_i is the height of the image.     
- h_o is the height of the object.     
- d_i is the image distance.     
- d_o is the object distance.     
Given:     
- The focal length, f= -12 cm (the focal length is negative for concave mirrors).     
- The image distance, d_i = -10 cm (the image distance is negative since the image is formed in front of the mirror).

• First, we need to find the object distance (d_o) using the mirror formula:

1/f=1/d_o + 1/d_i

• Substituting the known values:

1/-12 = 1/d_o + 1/-10

• Solving for 1/d_o

1/d_o= 1/-12 +  1/10 

= -5/60 + 6/60 = 1/60

Therefore, d_o = 60 cm

• Now we can find the magnification (M):     
  M= -d_i/d_o

= -(-10)/60 = 10/60 = 1/6

Therefore, the magnification of the concave mirror is 1/6. This means the image is 1/6th the size of the object and is inverted (as indicated by the negative sign in the magnification formula).

Q2. The radius of curvature of a convex mirror used for rear view on a bus is 4m. Determine the position of the image if a car is 2m away from the mirror.

Solution:

To determine the position of the image formed by a convex mirror, we use the mirror formula:

1/f = 1/v + 1/u

where:     
- f is the focal length of the mirror,     
- v is the image distance,     
- u is the object distance.

For a convex mirror, the focal length (f) is positive and is given by:

f = R/2

where (R) is the radius of curvature of the mirror.

Given:     
- The radius of curvature R = 4,     
- The object distance u = -2 cm (since the object distance is always taken as negative for real objects in mirror formulas).

• First, we calculate the focal length (f):

f = 4m/2 = 2m

• Next, we use the mirror formula to find the image distance (v):

1/f = 1/v + 1/u

• Substituting the known values:

½ = 1/v + 1/-2

• Solving for 1/v:

1/v = ½ + ½ 

• Rewriting and solving for (v):

1/v = ½ - ½ 

=½ + (-½) 

=½ - ½ = 0

Therefore, 1/v = 0; and v= ∞

This result indicates that the position of the image is at infinity, which implies that for an object placed at 2 meters from a convex mirror with a radius of curvature of 4 meters, the image appears to be very far behind the mirror, almost at an infinite distance.

In reality, due to the nature of a convex mirror, the image formed is virtual, erect, and diminished, located behind the mirror but virtually at a far distance, giving the appearance that it is at infinity.

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