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# Mirror Equation

## What Is Mirror Equation?

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

• The distance between the object and the pole of the mirror is called 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 the pole of the mirror is called Focal Length(f).

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

 $$\begin{array}{l}\frac{1}{v}+\frac{1}{u}=\frac{1}{f}\end{array}$$

Where,

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

The above formula is valid under all situations for all types of (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 visualization.

• For the measurement of all the distances, the optical center of the lens is considered.
• When the distances are measured opposite to the direction of the incident light, they are considered to be negative.
• When the distances are measured in the same direction of the incident light, they are considered to be positive.
• When the heights are measured upwards and perpendicular to the principal axis, they are considered to be positive.
• When the heights are measured downwards and perpendicular to the principal axis, they are considered to be negative.

### Mirror Equation for Concave Mirror and Mirror Equation for a Convex Mirror The mirror equation

$$\begin{array}{l}\frac{1}{v}+\frac{1}{u}=\frac{1}{f}\end{array}$$
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:

$$\begin{array}{l}f=\frac{R}{2}\end{array}$$
$$\begin{array}{l}\frac{1}{v}+\frac{1}{u}=\frac{1}{f}\end{array}$$

Calculation:

To calculate the Focal length of the given mirror, substitute the value of Radius of Curvature (R) in

$$\begin{array}{l}f=\frac{R}{2}\end{array}$$

We get-

$$\begin{array}{l}f=\frac{+4.00m}{2}=+2m\end{array}$$

Since,

$$\begin{array}{l}\frac{1}{v}+\frac{1}{u}=\frac{1}{f}\end{array}$$
we can re-arrange it as –

$$\begin{array}{l}\frac{1}{v}=\frac{1}{f}-\frac{1}{u}\end{array}$$

On substituting the values in the above equation, we get-

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

= 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.

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