A Detailed Breakdown Of The Mirror Formula

The world of light is fascinating and confusing. Objects in the mirror appear in different ways because of various phenomena. Reflection works differently for plane mirrors and spherical mirrors. 

Two kinds of spherical mirrors are convex and concave mirrors. Mirrors with reflecting surfaces with an inner curvature are concave mirrors, while those with outward curvature are convex mirrors. 

A mirror formula calculates the object’s distance from the mirror, the distance of image formed from mirrors, and any mirror’s focal length. Read ahead to find out more about spherical mirrors and mirror equations. 

The mirror formula holds for all kinds of spherical mirrors and the different positions at which an object can exist. Spherical mirrors -concave and convex are cut-out parts from whole spheres with their inner or outward surfaces polished, and other surfaces are shiny. 

The mirror formula calculates the positions of the image, object distance, and magnification. The procedure comes first, followed by sign conventions to minimise the chances of error. 

Mirror formula

The following equation is a representation of a commonly used mirror formula. 

1/v+1/u=1/f

• Here, f stands for the mirror’s focal length, 
• u stands for the object distance
• v stands for image distance from the mirror. 

Some commonly used terms when understanding the mirror formula-

• Pole (P)- The origin or the centre point of a mirror
• Centre of curvature (C) – The sphere’s centre point from which the mirror is cut. 
• The radius of curvature(r)- The sphere radius from which the mirror is cut. The radius is twice the focal length of the mirror. 
• Principal axis- A straight line passing through the pole(P) and centre of curvature (C )
• Principal Focus Point (F)- The point at which all parallel rays converge after reflection for a concave mirror and the point at which all parallel rays appear to diverge after reflection for a concave mirror.
• Focal length (f) is the distance measured from the mirror’s axis to its focal point.

Magnification

This mirror formula can also help in calculating the magnification of any mirror. Magnification is the ratio of the object’s height (h) to the height of the image(H₁). 

Therefore, Magnification is = h/H₁

When substituted with values of the distance of image (v) and distance of the object (u), we get 

m=-v/u

Here, the values of v and u follow the Cartesian sign conventions used. 

Sign Conventions Used 

It’s important to understand that sign conventions play an essential part in spherical mirror calculations. The New Cartesian sign convention is the most popular. Some of its rules followed are:

1. The mirror’s origin and pole(p) are at the same point. 
2. The x-axis of a coordinate system is the principal axis of any mirror. 
3. Any parallel distance from the central axis is measured from the axis or the origin. 
4. The object placed is always on the left side of the mirror. 
5. The light source is always on the left side of the mirror. 
6. The distance taken from the right-hand side of the mirror’s origin or pole (p) is positive, while those on the left-hand side are negative. The distance above the principal axis in a perpendicular fashion is positive, and the space below the central axis in a vertical manner is negative. 

Assumptions Used in Mirror Formula Derivation

• The number one assumption made is that object and image distance is measured from the origin of the mirror. 
• The second assumption is that the distances taken on the right-hand side of the mirror will be positive, while those on the left side will be negative. 
• The distances measured above the axis are assumed to be positive, and those below the axis are negative. 

Importance of The Mirror Formula

• The mirror formula calculates the exact position of an object and the distance image formed from the pole of the mirror. 
• It also calculates the magnifying effect of mirrors used for various daily life applications.
• To calculate simple problems associated with mirror distance placement and derive precise solutions. 

Examples

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

f = – 6cm

v = -3cm

1/u = 1/f-1/v

= -⅙ – (-⅓) 

1/u = ⅙

u = 6 cm

Thus, the position of the object is 6 cm from the mirror. 

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

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

= – 12.5

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

Conclusion

The mirror formula is widely used and has several applications. It’s essential to understand how to use it, the correct substitution of values, the correct procedure, and the accurate Cartesian sign system. When applied correctly, the mirror formula can help in solving all kinds of problems of mirrors and can simplify your work of lengthy calculations.