Table of Contents
Mirrors, Lenses and Image Formation
1. Introduction
Mirrors and lenses are important optical devices that control the direction and convergence or divergence of light rays to form images.
Mirrors form images through reflection, while lenses form images primarily through refraction.
They are used in spectacles, cameras, microscopes, telescopes, projectors, vehicle headlights and numerous scientific instruments.
The study of image formation through mirrors and lenses is an important part of geometrical optics.
2. Basic Optical Terminology
2.1 Principal Axis
The principal axis is the straight line passing through the centre of curvature and the pole of a spherical mirror, or through the optical centre and principal points of a simple thin lens.
2.2 Centre of Curvature
The centre of curvature is the centre of the sphere of which a spherical mirror forms a part.
It is represented by (C).
2.3 Radius of Curvature
The radius of curvature is the distance between the pole of a spherical mirror and its centre of curvature.
It is represented by (R).
2.4 Principal Focus
The principal focus is the point on the principal axis where rays parallel to the principal axis converge after reflection or refraction, or appear to diverge from.
It is represented by (F).
2.5 Focal Length
The focal length is the distance between the principal focus and the pole of a mirror or the relevant principal point of a thin lens.
It is represented by (f).
For a spherical mirror under the paraxial approximation:
[
f=\frac{R}{2}
]
2.6 Optical Centre
The optical centre of a thin lens is a point near its centre through which a ray passes approximately undeviated in the thin-lens approximation.
2.7 Aperture
The aperture is the effective diameter of the opening through which light enters or is reflected by an optical system.
It influences the amount of light collected and can affect image brightness and resolution.
3. Types of Mirrors
Mirrors are commonly classified into plane and spherical mirrors.
Spherical mirrors are divided into concave and convex mirrors.
3.1 Plane Mirror
A plane mirror has a flat reflecting surface.
Its image is:
- Virtual
- Erect
- Equal in size to the object
- Laterally inverted
- Located as far behind the mirror as the object is in front
3.2 Concave Mirror
A concave mirror has a reflecting surface curved inward.
It is also called a converging mirror because parallel rays close to the principal axis converge towards its principal focus after reflection.
Applications include:
- Reflecting telescopes
- Shaving mirrors
- Dentist’s mirrors
- Solar concentrators
- Reflectors in certain lighting systems
3.3 Convex Mirror
A convex mirror has a reflecting surface curved outward.
It is called a diverging mirror because parallel incident rays spread out after reflection and appear to originate from a virtual focus behind the mirror.
Its image is always:
- Virtual
- Erect
- Diminished
Applications include:
- Vehicle rear-view and side-view mirrors
- Security mirrors
- Mirrors used in shops and parking areas
Convex mirrors provide a wider field of view than plane mirrors.
4. Image Formation by Concave Mirrors
The nature and position of an image formed by a concave mirror depend on the object’s location.
| Object position | Image position | Nature of image |
|---|---|---|
| At infinity | At principal focus | Real, inverted, highly diminished |
| Beyond centre of curvature | Between C and F | Real, inverted, diminished |
| At centre of curvature | At C | Real, inverted, same size |
| Between C and F | Beyond C | Real, inverted, enlarged |
| At F | At infinity | Real, inverted, highly enlarged in the ideal model |
| Between F and pole | Behind the mirror | Virtual, erect, enlarged |
A real image can be formed on a screen. A virtual image cannot be projected onto a screen directly.
5. Image Formation by Convex Mirrors
A convex mirror always forms a virtual, erect and diminished image.
When the object is at infinity, the image forms at the principal focus behind the mirror.
For an object at a finite distance, the image forms between the pole and the principal focus behind the mirror.
This property makes convex mirrors useful where a wide field of view is required.
6. Mirror Formula
The mirror formula relates object distance, image distance and focal length.
Using the Cartesian sign convention:
[
\frac{1}{f}=\frac{1}{v}+\frac{1}{u}
]
Where:
- (f) = Focal length
- (v) = Image distance
- (u) = Object distance
For spherical mirrors:
- Distances measured in the direction of incident light are positive.
- Distances measured opposite to the direction of incident light are negative.
- Heights above the principal axis are positive.
- Heights below the principal axis are negative.
For a real object placed in front of a mirror, (u) is generally negative under this convention.
A concave mirror has negative focal length, while a convex mirror has positive focal length.
7. Magnification of Mirrors
Linear magnification is the ratio of image height to object height.
[
m=\frac{h_i}{h_o}
]
For spherical mirrors:
[
m=-\frac{v}{u}
]
Where:
- (m) = Magnification
- (h_i) = Image height
- (h_o) = Object height
- (v) = Image distance
- (u) = Object distance
Interpretation:
- Positive magnification indicates an erect image.
- Negative magnification indicates an inverted image.
- Magnification greater than 1 in absolute value indicates an enlarged image.
- Magnification less than 1 in absolute value indicates a diminished image.
8. Types of Lenses
A lens is a transparent optical element that refracts light to form an image.
Lenses are broadly classified as convex and concave.
8.1 Convex Lens
A convex lens is thicker at the centre than at the edges.
It is also called a converging lens because parallel rays of light generally converge towards a principal focus after passing through it.
Types include:
- Double convex lens
- Plano-convex lens
- Concavo-convex lens
Applications include:
- Magnifying glasses
- Cameras
- Microscopes
- Telescopes
- Spectacles for hypermetropia
8.2 Concave Lens
A concave lens is thinner at the centre than at the edges.
It is also called a diverging lens because parallel incident rays spread out after passing through it and appear to originate from a virtual focus.
Types include:
- Double concave lens
- Plano-concave lens
- Convexo-concave lens
Applications include:
- Spectacles for myopia
- Certain optical instruments
- Beam expansion systems
- Laser optics
9. Image Formation by Convex Lenses
The position and nature of the image depend on the location of the object.
| Object position | Image position | Nature of image |
|---|---|---|
| At infinity | At F on the opposite side | Real, inverted, highly diminished |
| Beyond 2F | Between F and 2F | Real, inverted, diminished |
| At 2F | At 2F on opposite side | Real, inverted, same size |
| Between F and 2F | Beyond 2F | Real, inverted, enlarged |
| At F | At infinity | Real, inverted, highly enlarged in the ideal model |
| Between F and optical centre | On the same side as object | Virtual, erect, enlarged |
A convex lens can produce either real or virtual images depending on the object’s position.
10. Image Formation by Concave Lenses
A concave lens always forms a virtual, erect and diminished image for a real object.
The image is formed on the same side of the lens as the object, between the optical centre and the principal focus.
This property makes concave lenses useful for correcting myopia.
11. Thin Lens Formula
The thin lens formula relates object distance, image distance and focal length.
Using the Cartesian sign convention:
[
\frac{1}{f}=\frac{1}{v}-\frac{1}{u}
]
Where:
- (f) = Focal length
- (v) = Image distance
- (u) = Object distance
For a real object placed to the left of a lens:
- Object distance (u) is negative.
- Image distance (v) is positive for a real image on the opposite side.
- Image distance (v) is negative for a virtual image on the same side.
- Focal length is positive for a converging convex lens.
- Focal length is negative for a diverging concave lens.
The sign convention must be applied consistently when solving numerical problems.
12. Magnification of Lenses
The linear magnification of a thin lens is:
[
m=\frac{h_i}{h_o}=\frac{v}{u}
]
Where:
- (h_i) = Image height
- (h_o) = Object height
- (v) = Image distance
- (u) = Object distance
A positive value indicates an erect image, while a negative value indicates an inverted image.
13. Power of a Lens
The power of a lens measures its ability to converge or diverge light.
It is defined as the reciprocal of focal length in metres.
[
P=\frac{1}{f}
]
The SI-derived practical unit is the dioptre (D).
If focal length is expressed in metres:
[
1\ D=1\ m^{-1}
]
Examples:
- A convex lens with focal length (0.5\ m) has power (+2\ D).
- A concave lens with focal length (-0.5\ m) has power (-2\ D).
A positive power indicates a converging lens, while a negative power indicates a diverging lens.
13.1 Combination of Thin Lenses
When thin lenses are placed in contact, their equivalent power is approximately the sum of their individual powers.
[
P=P_1+P_2+P_3+\cdots
]
This relationship assumes thin lenses in close contact and is commonly used in spectacle prescriptions.
14. Ray Diagrams
Ray diagrams are used to determine image position, size and nature.
14.1 Principal Rays for Mirrors
For a concave mirror:
- A ray parallel to the principal axis reflects through the principal focus.
- A ray passing through the principal focus reflects parallel to the principal axis.
- A ray passing through the centre of curvature retraces its path.
For a convex mirror, reflected rays diverge and their backward extensions pass through the virtual principal focus.
14.2 Principal Rays for Lenses
For a convex lens:
- A ray parallel to the principal axis passes through the principal focus on the opposite side.
- A ray passing through the optical centre travels approximately undeviated.
- A ray passing through the principal focus on the object side emerges parallel to the principal axis.
For a concave lens, a parallel ray emerges diverging as though it originated from the principal focus on the same side.
15. Applications of Mirrors and Lenses
15.1 Vehicle Mirrors
Convex mirrors are used to provide a wider field of view, helping drivers observe a larger area behind and beside a vehicle.
15.2 Shaving and Dentist’s Mirrors
Concave mirrors can produce enlarged, erect virtual images when the object is placed within their focal length.
15.3 Cameras
Camera lenses converge light to form real images on a sensor or photographic film.
15.4 Magnifying Glasses
A convex lens produces an enlarged, erect virtual image when the object is placed within its focal length.
15.5 Projectors
Projectors use converging optical systems to produce enlarged real images on a screen.
15.6 Telescopes
Telescopes use lenses or mirrors to collect and focus light from distant objects.
15.7 Microscopes
Microscopes use optical systems to magnify small objects that cannot be examined clearly with the unaided eye.
16. Worked Numerical Examples
Example 1: Concave Mirror
An object is placed 30 cm in front of a concave mirror with a focal length of 15 cm. Find the image distance.
Using the Cartesian sign convention:
[
u=-30\ cm
]
[
f=-15\ cm
]
Mirror formula:
[
\frac{1}{f}=\frac{1}{v}+\frac{1}{u}
]
[
-\frac{1}{15}=\frac{1}{v}-\frac{1}{30}
]
[
\frac{1}{v}=-\frac{1}{15}+\frac{1}{30}
]
[
\frac{1}{v}=-\frac{1}{30}
]
[
\boxed{v=-30\ cm}
]
The image forms 30 cm in front of the mirror. It is real, inverted and equal in size to the object.
Example 2: Convex Lens
An object is placed 30 cm from a convex lens of focal length 20 cm. Find the image distance.
Given:
[
u=-30\ cm
]
[
f=+20\ cm
]
Using:
[
\frac{1}{f}=\frac{1}{v}-\frac{1}{u}
]
[
\frac{1}{20}=\frac{1}{v}+\frac{1}{30}
]
[
\frac{1}{v}=\frac{1}{20}-\frac{1}{30}
]
[
\frac{1}{v}=\frac{1}{60}
]
[
\boxed{v=+60\ cm}
]
The image forms 60 cm on the opposite side of the lens. It is real and inverted.
Magnification:
[
m=\frac{v}{u}
]
[
m=\frac{60}{-30}=-2
]
The image is twice the size of the object and inverted.
Example 3: Power of a Lens
A convex lens has a focal length of 25 cm. Calculate its power.
Convert focal length into metres:
[
f=0.25\ m
]
[
P=\frac{1}{f}
]
[
P=\frac{1}{0.25}
]
[
\boxed{P=+4\ D}
]
17. Common Examination Misconceptions
- Concave mirrors are converging mirrors.
- Convex mirrors are diverging mirrors.
- Convex lenses are converging lenses.
- Concave lenses are diverging lenses.
- A convex mirror always forms a virtual, erect and diminished image.
- A concave lens always forms a virtual, erect and diminished image for a real object.
- A concave mirror can form real or virtual images.
- A convex lens can form real or virtual images.
- Real images can be projected onto a screen.
- Virtual images cannot be projected onto a screen directly.
- The focal length of a spherical mirror is half its radius of curvature.
- Lens power is measured in dioptres.
- The sign convention is essential for solving mirror and lens problems correctly.
36 Key Takeaways for Exams
- Mirrors form images through reflection.
- Lenses form images primarily through refraction.
- A plane mirror has a flat reflecting surface.
- A plane mirror forms a virtual, erect image equal in size to the object.
- A concave mirror is a converging mirror.
- A convex mirror is a diverging mirror.
- A concave mirror can form real or virtual images.
- A convex mirror always forms a virtual, erect and diminished image for a real object.
- The centre of curvature is the centre of the sphere of which a spherical mirror forms a part.
- The radius of curvature is the distance between the pole and centre of curvature.
- For a spherical mirror, (f=R/2) under the paraxial approximation.
- The mirror formula is (1/f=1/v+1/u).
- Mirror magnification is (m=-v/u).
- A convex lens is a converging lens.
- A concave lens is a diverging lens.
- A convex lens can form real or virtual images.
- A concave lens always forms a virtual, erect and diminished image for a real object.
- A real image can be formed on a screen.
- A virtual image cannot be projected onto a screen directly.
- The thin lens formula is (1/f=1/v-1/u).
- Lens magnification is (m=v/u).
- Lens power is the reciprocal of focal length in metres.
- The unit of lens power is the dioptre.
- A convex lens has positive power under the standard Cartesian convention.
- A concave lens has negative power under the standard Cartesian convention.
- A magnifying glass uses a convex lens.
- Vehicle rear-view mirrors commonly use convex mirrors.
- Concave mirrors can produce enlarged virtual images when objects are within the focal length.
- Cameras use lenses to form real images on sensors.
- Projectors produce enlarged real images on screens.
- Telescopes collect and focus light from distant objects.
- Microscopes magnify small objects.
- The principal axis is the reference line for geometrical image construction.
- The principal focus is where parallel rays converge or appear to diverge from.
- Ray diagrams help determine image position, orientation and size.
- Correct application of sign conventions is essential in numerical problems involving mirrors and lenses.