Mirror Formula and Magnification

Let us do a quick activity. Stand in front of a mirror and mark your position with a colored tape and label it as point A. Now, from point A, walk a little away from the mirror and mark it again with a colored tape and label it as point B. Can you calculate the distance from point A to point B? Need help in solving this problem? This distance can easily be calculated using the mirror formula. Let’s scroll ahead to find more.

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Sign Conventions

The sign convention for spherical mirrors follows a set of rules known as the “New Cartesian Sign Convention”, as mentioned below:

a. The pole (p) of the mirror is taken as the origin.

b. The principal axis is taken as the x-axis of our coordinate system.

c. The object is always placed on the left side of the mirror which implies that light falling from the object on the mirror is on the left-hand side.

d. All the distances parallel to the principal axis are measured from the pole (p) of the mirror.

e. All the distances measured from the pole (p) on the right-hand side of the mirror are taken as positive and those on the left-hand side of the mirror are taken as negative.

f. Distances measured perpendicular to and above the principal axis are taken as positive.

g. All the distances below the principal axis are taken as negative.

Learn Reflection of Light by Plane Mirror here

Mirror Formula

Now, that all these conventions are clear; let us know move on the mirror formula. Mirror Formula helps us to find:

a. Image distance which is represented as ‘v’.

b. Object distance which is represented as ‘u’.

c. Focal length which is represented as ‘f’.

And is written as :

\( \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \)

This formula is valid for all kinds of spherical mirrors, for all positions of the object. Although one needs to be careful about the values, one puts for u,v and f with appropriate sign according to the sign convention given above.

Learn about Terminology of Spherical Mirrors and its types here

Magnification 

Physically, we all understand what is magnification. It can be defined as the extent to which the image appears bigger or smaller in comparison to the object size.

It is represented as the ratio of the height of the image to the ratio of the height of the object. Magnification is denoted as the letter ‘m’. Where,

Magnification (m) = h/h’ 

And h’ is the image height and h is the object height.

Magnification can also be related to the image distance and object distance; therefore it can also be written as:

m = -v/u

Where v is the image distance and u is the object distance.

Hence, the expression for magnification (m) becomes:

m = h’/h = -v/u

Learn more about Reflection of Light here

Solved Example for You

Q. What will be the distance of the object, when a concave mirror produces an image of magnification m? The focal length of the mirror is f.

a. \( \frac{1}{m(m+1)}\)                                   b. (m-1)f

c. \( \frac{f}{m(m-1)}\)                                   d. (m+1)f

Sol: (c.) \( \frac{f}{m(m-1)}\)

Given, m = -v/u

=> v = -mu

By mirror formula,

1/f = 1/v + 1/u = 1/-(mu) + 1/u

=> 1/f = 1/u ( -1/m + 1 )      OR     u =\( \frac{f}{m(m-1)}\)