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.
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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 \(\frac{1}{v}+\frac{1}{u}=\frac{1}{f}\) 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 4.00Â 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.
