6.3 Image Formation by Lenses
A lens is a piece of transparent material, such as glass, perspex, or plastic, having two surfaces with at least one curved surface. The two main types of lenses are convex lenses and concave lenses. A convex lens acts as a converging lens, while a concave lens acts as a diverging lens. Understanding the paths of light rays through lenses allows us to determine image positions, their characteristics, and calculate linear magnification.
Learning Objectives
- Explain the converging action of a convex lens and the diverging action of a concave lens.
- Define and identify optical terms: optical centre (\(O\)), principal axis, axis of lens, focal point (\(F\)), object distance (\(u\)), image distance (\(v\)), and focal length (\(f\)).
- Describe how to estimate the focal length of a convex lens using a distant object.
- Construct ray diagrams using three primary rays to determine the position and characteristics of images formed by convex and concave lenses for various object distances.
- Define linear magnification, \(m\), and calculate its value using the ratio of image height to object height or image distance to object distance.
Summary of Light Ray Refraction and Image Characteristics for Convex and Concave Lenses
Compare how parallel light rays refract through both lens types, identify real and virtual focal points, and review the three construction rays used in ray diagrams.
Convex Lens — Converging Action
Concave Lens — Diverging Action
Short Explanation
Types of Lenses and Their Actions
A convex lens is a converging lens. Parallel light rays passing through it converge at a real focal point, \(F\), so its focal length, \(f\), is positive. A concave lens is a diverging lens. Parallel light rays appear to diverge from a virtual focal point, \(F\), so its focal length, \(f\), is negative.
Optical Terms
The optical centre, \(O\) is the point at the centre of the lens. The principal axis passes through \(O\) and the centre of curvature of both surfaces. The focal point, \(F\) lies on the principal axis. Object distance, \(u\), image distance, \(v\), and focal length, \(f\) are measured from \(O\).
Estimating Focal Length
Direct a convex lens towards a distant object. Move a white screen behind the lens until a sharp, real, inverted image is focused. The distance between the optical centre and the screen gives the estimated focal length, \(f\). This method is not suitable for a concave lens because it produces only virtual images, which cannot be captured on a screen.
Linear Magnification
Linear magnification, \(m\) is the ratio of image height, \(h_i\), to object height, \(h_o\), and also equals the ratio of image distance, \(v\), to object distance, \(u\). It has no units.
Linear magnification
\(m = \dfrac{h_i}{h_o} = \dfrac{v}{u}\)\(m < 1\): diminished. \(m = 1\): same size. \(m > 1\): magnified.
| Object position for convex lens | Image position | Characteristics |
|---|---|---|
| \(u > 2f\) | Between \(F\) and \(2F\) (\(f < v < 2f\)) | Real, inverted, diminished |
| \(u = 2f\) | At \(2F\) (\(v = 2f\)) | Real, inverted, same size as object |
| \(f < u < 2f\) | Beyond \(2F\) (\(v > 2f\)) | Real, inverted, magnified |
| \(u = f\) | At infinity | Virtual, upright, magnified |
| \(u < f\) | Behind object on the same side (\(v > u\)) | Virtual, upright, magnified |
| Object at infinity | At \(F\) (\(v = f\)) | Real, inverted, diminished |
| Concave lens object position | Image position | Characteristics |
|---|---|---|
| Any object distance | Between optical centre and focal point (\(v < f\)) | Virtual, upright, diminished |
Try to Answer First
Answer in your mind, then press “Check Answer”.
Why is the focal length of a concave lens considered negative?
Check Answer
An object is placed at a distance of \(15\text{ cm}\) in front of a convex lens with a focal length of \(10\text{ cm}\). State the characteristics of the image formed.
Check Answer
What are the characteristics of an image formed by a concave lens for any object distance?
Check Answer
Common Mistakes
- Confusing the focal point characteristics: thinking concave lenses form real focal points because light passes through them. In reality, the rays diverge, making the focal point virtual.
- Incorrectly drawing the ray passing through the optical centre (\(O\)): adding refraction or bending at \(O\), whereas rays passing through the optical centre travel in a straight line without being refracted.
- Stating units for linear magnification: students often mistakenly write units such as \(\text{cm}\) for \(m\), but linear magnification is a dimensionless ratio.
- Believing convex lenses always magnify: convex lenses can also produce diminished or same-sized images depending on whether \(u > 2f\) or \(u = 2f\).
Concept Misunderstandings
Real images can be formed on the same side of the lens as the object.
For lenses, real images are formed by intersecting rays on the opposite side of the lens and can be captured on a screen. Virtual images are formed on the same side as the object and cannot be caught on a screen.
A concave lens can form a magnified image if the object is placed very close to it.
Concave lenses always produce virtual, upright, and diminished images regardless of object position.
Estimating the focal length using a distant object on a screen works for both lenses.
This method only works for converging (convex) lenses because concave lenses do not form real images on a screen.
Summary
- A convex lens converges parallel rays to a real focus (\(f\) is positive).
- A concave lens diverges parallel rays so they appear to emerge from a virtual focus (\(f\) is negative).
- The three construction rays pass through the optical centre undeflected, travel parallel then through/from \(F\), or travel through/towards \(F\) then emerge parallel.
- Convex lens images vary: real and inverted when \(u > f\); virtual, upright, and magnified when \(u < f\).
- Concave lens images are always virtual, upright, and diminished (\(v < f\)).
- Linear magnification is given by \(m = \dfrac{h_i}{h_o} = \dfrac{v}{u}\) and has no unit.
Short Activity
Objective: Identify lens actions, image characteristics, optical terms and linear magnification.
A. Objective Quiz
1 Which of the following statements correctly describes a concave lens?
2 A convex lens with a focal length of \(10\text{ cm}\) produces a sharp image on a screen placed \(20\text{ cm}\) behind the lens. What is the linear magnification of this image?
B. Fill in the Blanks
3 Light rays that pass straight through the of a lens travel in a straight line without being refracted.
4 The formula for linear magnification in terms of image distance \(v\) and object distance \(u\) is \(m =\) .
C. Matching / Drag and Drop
Match each object position for a convex lens to the correct characteristics of the image formed. If using a phone, tap the answer first, then tap the matching box.
