Mirror Equation: Sign Conventions, Derivations, Examples & Uses

Mirror equations and formulas help us to predict the position where the image will be formed if we know the object position and the focal length of the mirror. Thus, in the case of a real image, if we place a screen at that position, then we will get the image of the object on the screen; else, to capture the image, we need to move the screen at probable positions and check. In this article, we will study the mirror equation in detail.

What is Mirror Equation?

Mirror Equation is the relation that relates the object distance, the image distance, and the focal length of the mirror with each other. The object distance is the distance of the object from the reflecting surface of the mirror. It is denoted by the symbol \(u.\)

The image distance is the distance of the image from the reflecting surface of the mirror. It is denoted by the symbol \(v.\) For small apertures, the focal length of the mirror is the distance between its pole and principal focus. Focal length is denoted by the symbol \(f.\)

The focal length \(f.\) of the mirror is related to its radius of curvature \(\left( R \right)\) by the relation:
\(f = \frac{R}{2}\)
Thus, the mirror equation is given by the relation:
\(\frac{1}{v} + \frac{1}{u} = \frac{2}{R} = \frac{1}{f}\)

Sign Conventions for Mirror Equation

Mirror Equation follows certain sign conventions that are as under:

1. The principal axis of the mirror is taken along the \(x\)-axis of the rectangular coordinate system, and its pole is taken as the origin.
2. The object is taken on the left side of the mirror; that is, light is incident on the mirror from the left-hand side.
3. All the distances parallel to the principal axis of the mirror are measured from the pole of the mirror.
4. The distances measured in the direction of the incident light are taken as positive.
5. The distances measured in the direction opposite to the direction of incident light are taken as negative.
6. The heights measured upwards and perpendicular to the principal axis of the mirror are taken as positive.
7. The heights measured downwards and perpendicular to the principal axis of the mirror are taken as negative. These sign conventions can be summarized in the below figure:

Thus, the focal length of a concave mirror is taken negative, and the focal length of a convex mirror is taken as positive. Also, the object distance and the object height are taken as negative and positive, respectively.

Derivation of Mirror Equation for a Concave Mirror

The Mirror Equation for a concave mirror can be derived using the ray diagram as shown below:

In the above ray diagram, the object \(AB\) is placed on the principal axis of the concave mirror beyond its centre of curvature \(\left( C \right)\) that forms an image \(A’B’\) between the centre of curvature \(\left(C \right)\) and principal focus \(\left( F \right)\) of the concave mirror. Here, we have taken the case where a real image is formed by a concave mirror.

As shown in the ray diagram, \(\Delta ABC\) and \(\Delta A’B’C\) are similar triangles.
\(\therefore \frac{{AB}}{{A’B’}} = \frac{{CB}}{{CB’}}\)
Similarly, \(\Delta ABP,\) and \(\Delta A’B’P\) are also similar triangles
\(\therefore \frac{{AB}}{{A’B’}} = \frac{{PB}}{{PB’}}\)
Thus, combining both the relations we get,
\(\frac{{AB}}{{A’B’}} = \frac{{CB}}{{CB’}} = \frac{{PB}}{{PB’}}\)
\(\therefore \frac{{CB}}{{CB’}} = \frac{{PB}}{{PB’}}\)
Measuring all the distances from the pole \(\left( P \right)\) we get,
\(CB = PB – PC\)
\(CB’ = PC – PB’\)
Putting the values of \(CB\) and \(CB’\) in the above equation, we get,
\(\frac{{PB – PC}}{{PC – PB’}} = \frac{{PB}}{{PB’}}\)
Using cartesian sign convention, we get,
\(PB = – u\)
\(PC = – R\)
\(PB’ = – v\)
Here, \(u\) is the object distance, \(v\) is the image distance, and \(R\) is the radius of curvature.

Now, substituting the values of \(PB,PB’\) and \(PC\) in the above equation, we get,
\(\frac{{ – u – \left({ – R} \right)}}{{ – R – \left({ – v} \right)}} = \frac{{ – u}}{{ – v}}\)
\(\therefore \frac{{ – u + R}}{{ – R + v}} = \frac{u}{v}\)
Now, by simplifying the equation, we get,
\(v\left({ – u + R} \right) = u\left({ – R + v} \right)\)
\( – vu + vR = – uR + uv\)
or
\(uR + vR = 2uv\)
Now, dividing both the sides by \(uvR\), we get,
\(\frac{1}{v} + \frac{1}{u} = \frac{2}{R}\)
But we know that, \(f = \frac{R}{2}\)
\(\therefore \frac{1}{v} + \frac{1}{u} = \frac{1}{f}\) which is the required mirror equation. This equation holds good even when a concave mirror forms a virtual image.
Thus, the mirror equation for a concave mirror is \(\frac{1}{v} + \frac{1}{u} = \frac{1}{f}.\)

Derivation of Mirror Equation for a Convex Mirror

Mirror Equation for a convex mirror can be derived using the ray diagram as shown below:

In the above ray diagram, the object \(AB\) is placed on the principal axis of the convex mirror that forms an image \(A’B’\) between the principal focus \(\left( F \right)\) and pole \(\left(P \right)\) of the convex mirror. A convex mirror always forms a virtual and erect image of the object, whatever may be its position.

As shown in the ray diagram, \(\Delta ABC\) and \(\Delta A’B’C\) are similar triangles.
\(\therefore \frac{{AB}}{{A’B’}} = \frac{{CB}}{{CB’}}\)
Similarly, \(\Delta ABP,\) and \(\Delta A’B’P\) are also similar triangles
\(\therefore \frac{{AB}}{{A’B’}} = \frac{{PB}}{{PB’}}\)
Thus, combining both the relations we get,
\(\frac{{AB}}{{A’B’}} = \frac{{CB}}{{CB’}} = \frac{{PB}}{{PB’}}\)
\(\therefore \frac{{CB}}{{CB’}} = \frac{{PB}}{{PB’}}\)
Measuring all the distances from the pole \(\left(P \right),\) we get,
\(CB = PB + PC\)
\(CB’ = PC – PB’\)
Putting the values of \(CB\) and \(CB’\) in the above equation, we get,
\(\frac{{PB + PC}}{{PC – PB’}} = \frac{{PB}}{{PB’}}\)
Using cartesian sign convention, we get,
\(PB = – u\)
\(PC = + R\)
\(PB’ = + v\)
Here, \(u\) is the object distance, \(v\) is the image distance, and \(R\) is the radius of curvature.

Now, substituting the values of \(PB,PB’\) and \(PC\) in the above equation, we get,
\(\frac{{ – u + R}}{{R – v}} = \frac{{ – u}}{v}\)
\(\therefore \frac{{ – u + R}}{{ – R + v}} = \frac{u}{v}\)
Now, by simplifying the equation, we get,
\(v\left({ – u + R} \right) = – u\left({R – v} \right)\)
\(- vu + vR = – uR + uv\)
or
\(uR + vR = 2uv\)
Now, dividing both the sides by \(uvR,\) we get,
\(\frac{1}{v} + \frac{1}{u} = \frac{2}{R}\)
But we know that, \(f = \frac{R}{2}\)
\(\therefore \frac{1}{v} + \frac{1}{u} = \frac{1}{f}\)
which is the required mirror equation.
Thus, the mirror equation for a convex mirror is \(\frac{1}{v} + \frac{1}{u} = \frac{1}{f}.\)
Hence, the mirror equation, that is, \(\frac{1}{v} + \frac{1}{u} = \frac{1}{f}\) is the same for both the concave and the convex mirrors. And this equation remains unaffected whether the image formed is real or virtual.

Solved Examples – Mirror Equation

Q.1. An object is kept \(60\,{\text{cm}}\) in front of a concave mirror of focal length \(30\,{\text{cm}}{\text{.}}\) Find the position of the image formed.
Ans:
Given that,
The object distance is \(u = – 60\,{\text{cm}}\)
The focal length of the concave mirror is \(f = – 30\,{\text{cm}}\)
The image distance is given by the relation,
\(\frac{1}{v} = \frac{1}{f} – \frac{1}{u} = \frac{1}{{ – 30}} – \frac{1}{{ – 60}}\)
\(\therefore \frac{1}{v} = \frac{{ – 1}}{{30}} + \frac{1}{{60}} = \frac{{ – 2 + 1}}{{60}} = \frac{{ – 1}}{{60}}\)
\(\therefore v = – 60\,{\text{cm}}\)
Hence, the image is formed in front of the concave mirror at a distance of \(60\,{\text{cm}}.\)

Q.2. A convex mirror used for rearview on the bus has a radius of curvature of \( {\text{4}}\,{\text{m}}.\) If a car is located at \(2\,{\text{m}}\) from the mirror, find the position of the image.
Ans:
Given that,
The object distance is \(u = – 2\,{\text{m}}\)
The radius of curvature of the convex mirror is \(R = 4\,{\text{m}}\)
So, the focal length of the convex mirror is \(f = \frac{R}{2} = \frac{4}{2} = 2\,{\text{m}}\)
The image distance is given by the relation,
\(\frac{1}{v} = \frac{1}{f} – \frac{1}{u} = \frac{1}{2} – \frac{1}{{ – 2}}\)
\(\therefore \frac{1}{v} = \frac{1}{2} + \frac{1}{2} = \frac{2}{2} = \frac{1}{1}\)
\(\therefore v = 1\,{\text{m}}\)
Hence, the image of the car is formed behind the mirror at a distance of \(1\,{\text{m}}{\text{.}}\)

Applications of Mirror Equation

Mirror Equation finds its application in the following ways:

1. Mirror equation helps in predicting the image distance when the object distance and the focal length of the mirror are known.
2. The equation helps in finding the object distance when the image distance and the focal length of the mirror are known to us.
3. Mirror equation helps us in calculating the focal length of the mirror just by knowing the distance of the object and the distance of the image it forms.
4. Using the mirror equation along with the magnification equation helps us in getting the value of either the image height or the object height when the other one is given.

Summary

Hope you understood the mirror equation in detail along with the sign conventions used and its derivations using a concave mirror as well as a convex mirror. This article would also have helped you understand the usage of mirror equations in solving numerical problems and applications of mirror equations.

FAQs

Q.1. What kind of images do a convex mirror form?
Ans: A convex mirror forms only virtual images that are erect and diminished.

Q.2. How can you distinguish whether a mirror is a convex mirror or a concave mirror just by looking at the virtual image formed by it?
Ans: A concave mirror always forms virtual and enlarged images, whereas a convex mirror forms virtual and diminished images. So, if the virtual image formed by the mirror is enlarged, then it is a concave mirror; else, if the image formed is diminished, then the mirror is a convex mirror.

Q.3. What is the mirror equation?
Ans: The equation \(\frac{1}{v} + \frac{1}{u} = \frac{1}{f}\) is the mirror equation that relates the object distance, the image distance and the focal length of the mirror.

Q.4. What is the sign used for the focal length of a concave mirror?
Ans: A negative sign is used for the focal length of a concave mirror.

Q.5. What is the sign used for the focal length of a convex mirror?
Ans: A positive sign is used for the focal length of a concave mirror

Q.6. What is the focal length of the mirror?
Ans: The distance between the pole and the principal focus of a mirror is called its focal length.

Q.7. What kind of images do a concave mirror form?
Ans: A concave mirror forms both real and virtual images. It can form enlarged, same size or diminished real images but can form only enlarged virtual images.

We hope you find this article on Mirror Equation helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.