Mirror Formula for Spherical Mirrors

Introduction

Light is electromagnetic radiation and is the only part of the electromagnetic spectrum that can be visible to the human eye. If the light illuminates any surface the quantity of light that is received by the surface depends on the light source’s distance from the surface. The law of illumination or inverse square law of illumination gives the relation between the source and the surface. It explains the variation of illumination of the surface with the distance squared tween the source and the surface. When light falls on the rough surface it can be seen in all directions. But for a polished surface, the proper illumination is seen at a particular angle because of the reflection of the light.

What is a mirror?

A mirror is an object which is used to see the images of objects and living things. If the object is placed ahead of the mirror the light is reflected from the mirror surface and makes the image of the object on the mirror. Mirrors act like water that shows the image of an object in front of it. This happens by spraying a thin layer of aluminium or silver at the back of the mirror. Because they have high reflectivity. Thus the light that falls on the mirror is reflected on the backside of the mirror.

Types of mirror

Mirrors are classified as plane mirrors and spherical mirrors according to their shapes.

Plane mirror

A plane mirror is a thin flat glass piece that is coated with metal at the backside of the glass. The image size and the object size are the same in the plane mirror. Both virtual and real images can be formed by plane mirrors. Several images of the object can be produced by the plane mirrors if the plane mirrors are held at a particular angle θ. This is the principle used in kaleidoscopes. The plane mirror has an infinite radius of curvature and focal length. The image is formed at a distance that is similar to the distance the object is placed from the plane mirror. The image from the plane mirror is erect, virtual, and inverted.

Image formation by plane mirror

Spherical mirror

These mirrors are highly polished surfaces that are cut as part of a sphere. One side of the spherical mirror is a reflecting surface and the other side is silvered. The middle part of the spherical mirror is the pole. The center of the spherical mirror is the center of curvature. The radius of the spherical mirror is the radius of curvature. The Principal axis of the spherical mirror is formed by the line that joins the pole and center of curvature. The focal length is the measured distance of the pole from the principal focus.

Types of spherical mirrors

The spherical mirrors are of two types: concave mirrors and convex mirrors.

Concave mirror

The spherical mirror in which the light is reflected from the bent surface of the mirror is known to be a concave mirror. Its outer side is coated with silver and the other side is polished. Concave mirrors can produce virtual and real images. The image size is characterized by the distance covered between the object and the mirror.

Convex mirror

A spherical mirror in which the light is reflected from the curved surface of the mirror or the bulged outward mirror surface is called a convex mirror. The other name for a convex mirror is a diverging mirror. The images formed by these mirrors are always virtual, diminished, and erect.

Derivation for mirror formula

The mirror formula or mirror equation gives the relation between the object and the image with the focal length. The mirror equation is suitable for both plane and spherical mirrors. The mirror formula is represented as

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

f denotes the mirror’s focal length. u denotes the image distance. v denotes the object’s distance.

The distance which is measured in the same way as the incident light is noted as positive and that calculated in the opposite way of the incident light is noted as negative. Let us consider an object AB in the principle axis of the concave mirror which produces a real image. The image is formed between the radius of curvature and focus.

$$\mathrm{let\: AB=h_o ,A'B'=h_i,FP=f,BP=u ,B'P=v and PC=R,}$$

$$\mathrm{From\: the\: assumption\: \mathit{\frac{h_o}{h_i} =\frac{u-2f}{2f-v}}……………………………….(1)}$$

$$\mathrm{Also, \mathit{\frac{h_o}{h_i} =\frac{f}{v-f}}………………………………….(2)}$$

$$\mathrm{From\: equations \:(1)\: \&\: (2)\:\:\: \mathit{\frac{f}{v-f}=\frac{u-2f}{2f-v}}}$$

$$\mathit{2f^2-vf=uv-2vf-uf+2f^2}$$

$$\mathit{uv-vf-uf=0}$$

$$\mathit{uv=uf+vf}$$

$$\mathrm{Divide\: the\: equation\: by \:uvf, }$$

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

Sign convention

• The object is placed on the left side of the mirror.

• The height that is measured above the principle axis is noted as positive and which is measured below is noted as negative.

• The distance that is measured on the right side of the pole is taken as positive and that measured on the left side is taken as negative.

• The distances are generally measured from the pole.

Conclusion

Whenever a light falls on the mirror surface it either reflected or refracted. The mode of refraction and reflection also depends on the types of the mirror. In this tutorial, the facts about mirrors, spherical mirrors, and types of mirrors were discussed. The derivation for mirror formula and sign convention was also discussed in detail.

FAQs

1. What are the laws of reflection?

The laws of reflections are given below.

• The rays of light that fall on the surface are incident light and the rays that jump off from the surface are called reflected light.

• An non-existing line that passes normal to the reflecting surface at the point of the incident is called normal.

• The incident angle is equal to the reflected angle.

2. What are the uses of concave mirrors?

The concave mirrors have a variety of applications. It is used in vehicle headlights, torches, flashbacks, searchlights, dental mirrors, ophthalmoscopes, solar furnaces, telescopes, microscopes, shaving mirrors, and makeup mirrors.

3. Differentiate regular reflection and diffused reflection.

Regular reflection	Diffused reflection
Smooth surfaces undergoes regular reflection	Rough surfaces undergoes diffused reflection
Reflected rays are regular.	Reflected rays are irregular.
Reflection can be seen.	Reflection cannot be seen.

4. What is magnification?

The proportion between the image size and the object size is stated as the linear magnification of the mirror. It is denoted as m.

$$\mathrm{Magnification (m)=\frac{image \:height}{object\: height}}$$

5. What are the uses of convex mirrors?

• It is used in the hallway of the buildings, and side mirrors of the vehicles.

• It is used in the curves and turns in the road and blind spots.