 # Mirror Equation

## What is Mirror Equation?

It is an equation relating object distance and image distance with focal length is known as a mirror equation. It is also known as mirror formula. In a spherical mirror:

• The distance between the object and the pole of the mirror is called the object distance(u).
• The distance between the image and the pole of the mirror is called Image distance(v).
• The distance between the Principal focus and pole of the mirror is called Focal Length(f).

In ray optics, The object distance, Image distance and Focal length are related as,

 $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

Where,

• u is the Object distance
• v is the Image distance
• f is the Focal Length given by $f=\frac{R}{2}$
• R is the radius of curvature of the spherical mirror

The above formula is valid under all situation for all types of spherical mirrors (Concave and Convex) and for all object positions.

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## Sign Conventions New Cartesian Sign Convention is used to avoid confusion in understanding the ray directions. Refer to the diagram for clear visualisation.

• All distances are measured considering mirror as the origin
• A convex mirror has a negative focal length, whereas the concave mirror has a positive focal length.
• Virtual distances are negative(-) whereas Real distances are positive(+)

### Mirror Equation for concave mirror and Mirror Equation for convex mirror The mirror equation $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$ holds good for concave mirrors as well as convex mirrors.

## Example of Mirror Equation

The radius of curvature of a convex mirror used for rearview on a car is 4.00m. If the location of the bus is 6 meter from this mirror, Find the position of the image formed.

Solution:

Given:

The radius of curvature (R)= +4.00m

Object distance(u) = -6.00m

Image distance(v) = ?

Formula used:

$f=\frac{R}{2}$ $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

Calculation:

To calculate the Focal length of the given mirror, Substitute the value of Radius of Curvature (R) in the $f=\frac{R}{2}$ . we get-

$f=\frac{+4.00m}{2}=+2m$

Since, $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$ we can re- arrange it as –

$\frac{1}{v}=\frac{1}{f}-\frac{1}{u}$

On substituting the values in the above equation we get-

$\Rightarrow\frac{1}{v} =\frac{1}{+2.00}-\frac{1}{\left ( -6.00 \right )}$ $=\frac{1}{2.00}+\frac{1}{6.00}$ $\frac{6+2}{2\times 6}=\frac{8}{12}$ $v=\frac{12}{8}$

= 1.5 meter.

The image is 1.5 meters behind the mirror.

Mirror is a polished surface which reflects the incident light to form the image. Here reflected light will have a wavelength and many other physical properties almost the same as that of the incident light.