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 a 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 situations for all types of spherical mirrors (Concave and Convex) and for all object positions.

You may also want to check out these topics given below!

Sign Conventions

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

• 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 a 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 4.00 m. If the location of the bus is 6 meters from this mirror, find the position of the image formed.

Solution:

Given:

The radius of curvature (R)= +4.00 m

Object distance(u) = -6.00 m

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.

The 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.