Light – Reflection and Refraction · Lesson 12 of 15
Image Formation by Lenses
“Move the object and the image begins a full relocation programme.”
• Describe all six convex-lens image cases. • Describe images formed by a concave lens. • Predict image position, size, orientation and nature. • Explain when a lens image can be obtained on a screen. • Relate object movement to image movement.
Slide a candle toward a convex lens while keeping a screen on the other side. The sharp image moves and grows, then disappears from the screen when the candle crosses F₁. The pattern resembles a concave mirror, but the real image forms on the opposite side of the lens.
Convex Lens
| Object position | Image position | Relative size | Nature |
|---|---|---|---|
| At infinity | At F₂ | Highly diminished, point-sized | Real and inverted |
| Beyond 2F₁ | Between F₂ and 2F₂ | Diminished | Real and inverted |
| At 2F₁ | At 2F₂ | Same size | Real and inverted |
| Between F₁ and 2F₁ | Beyond 2F₂ | Enlarged | Real and inverted |
| At F₁ | At infinity | No finite screen position | Emergent rays parallel |
| Between F₁ and O | Same side as object | Enlarged | Virtual and erect |
Outside F₁, the convex lens forms a real inverted image on the opposite side. At 2F₁ the image is same-sized at 2F₂. Moving the object toward F₁ enlarges the image and pushes it farther away. Inside F₁, refracted rays diverge; their backward extensions form a virtual erect enlarged image on the object's side.
Problem
Predict the image of a distant tree through a convex lens.
- 1.The rays are nearly parallel.
- 2.They converge near F₂.
- 3.The image is highly diminished, real and inverted, and can be obtained on a screen.
Detailed Ray Diagrams for the Six Convex-Lens Cases
The table above gives the result of each object position. The following constructions show how each image is formed. In every diagram, blue lines are incident rays, red lines are refracted rays, the orange arrow is the object and the purple arrow or point is the image. A solid line represents the actual path of light. Dashed purple lines are backward extensions used only to locate a virtual image. The symbols F₁ and 2F₁ lie on the object's side of the lens, F₂ and 2F₂ lie on the opposite side, and O is the optical centre.
Object at Infinity
An object at a very large distance sends rays that reach the convex lens almost parallel to the principal axis. After refraction, these rays converge at the second principal focus, F₂. Because the refracted rays actually meet, the image is real and can be obtained on a screen placed at F₂. Rays from different points of a distant object form a very small inverted image close to F₂. In the limiting case shown here, the image is represented by a point at F₂.
• Object position: At infinity. • Image position: At F₂. • Relative size: Highly diminished or point-sized. • Orientation: Inverted. • Nature: Real. • Screen test: The image can be obtained on a screen placed at F₂.
Object Beyond 2F₁
Place the object farther from the lens than 2F₁. From the top of the object, draw one ray parallel to the principal axis; after refraction it passes through F₂. Draw a second ray through the optical centre O; it continues in the same direction without appreciable deviation. The two refracted rays intersect between F₂ and 2F₂. Their actual intersection forms a real image. The image arrow lies below the principal axis, showing that it is inverted, and is shorter than the object arrow, showing that it is diminished.
• Object position: Beyond 2F₁. • Image position: Between F₂ and 2F₂. • Relative size: Diminished. • Orientation: Inverted. • Nature: Real. • Screen test: The image can be obtained on a screen.
Object at 2F₁
Place the object at 2F₁. A ray parallel to the principal axis refracts through F₂, while a ray through the optical centre O continues undeviated. The refracted rays meet at 2F₂. The image and object are equally far from the optical centre on opposite sides of the lens. Their arrows have equal height, but the image arrow points below the principal axis. Therefore, the image is real, inverted and the same size as the object. Its magnification is −1.
• Object position: At 2F₁. • Image position: At 2F₂. • Relative size: Same size as the object. • Orientation: Inverted. • Nature: Real. • Magnification: −1. • Screen test: The image can be obtained on a screen.
Object Between F₁ and 2F₁
Place the object between F₁ and 2F₁. Draw a ray parallel to the principal axis and refract it through F₂. Draw a second ray through the optical centre O without changing its direction. The rays meet beyond 2F₂. Since actual refracted rays meet, the image is real. It is below the principal axis and taller than the object, so it is inverted and enlarged. As the object moves from 2F₁ toward F₁, the image moves farther beyond 2F₂ and becomes progressively larger.
• Object position: Between F₁ and 2F₁. • Image position: Beyond 2F₂. • Relative size: Enlarged. • Orientation: Inverted. • Nature: Real. • Screen test: The image can be obtained on a screen.
Object at F₁
Place the object at the first principal focus, F₁. A ray parallel to the principal axis refracts through F₂. A second ray passes through the optical centre O without appreciable deviation. In this special position, the two refracted rays emerge parallel to each other and therefore do not meet at any finite distance. The image is described as forming at infinity. It is highly enlarged, real and inverted in the limiting theoretical sense, but no finite screen position can capture the complete image.
• Object position: At F₁. • Image position: At infinity. • Relative size: Highly enlarged. • Orientation: Inverted. • Nature: Real in the limiting theoretical sense. • Ray behaviour: Emergent rays are parallel. • Screen test: No finite screen position captures the complete image.
Object Between F₁ and O
Place the object between the first principal focus F₁ and optical centre O. A ray parallel to the principal axis refracts toward F₂. A second ray passes through O without appreciable deviation. The refracted rays spread apart on the opposite side of the lens and therefore cannot form a real image there. Extend both refracted rays backward on the object's side using dashed lines. Their extensions meet to locate a virtual image. The image arrow is upright and taller than the object arrow, so the image is erect and enlarged. Because actual light rays do not pass through the image position, the image cannot be obtained on a screen.
• Object position: Between F₁ and O. • Image position: On the same side of the lens as the object. • Relative size: Enlarged. • Orientation: Erect. • Nature: Virtual. • Screen test: The image cannot be obtained on a screen. • Motion: As the object approaches O, the virtual image moves toward the lens and becomes smaller while remaining enlarged.
Problem
Where should an object be placed to obtain a same-sized real image?
- 1.Place it at 2F₁.
- 2.A parallel ray and an optical-centre ray meet at 2F₂.
- 3.The image is real, inverted and has the same magnitude of height.
Problem
Why must print be held within F₁ when using a convex magnifier?
- 1.An erect enlarged image must be virtual.
- 2.A convex lens gives that result only between F₁ and O.
- 3.The backward extensions meet on the object's side, so the eye sees enlarged upright print.
Concave Lens
| Object position | Image position | Relative size | Nature |
|---|---|---|---|
| At infinity | At F₁ | Highly diminished, point-sized | Virtual and erect |
| Any finite position | Between F₁ and O | Diminished | Virtual and erect |
A concave lens always makes incident rays diverge. For a real object, only backward extensions can meet, so the image is always virtual and on the object's side. It remains erect and smaller than the object. Moving the object far away moves the image toward F₁ and makes it smaller.
Detailed Ray Diagrams for the Two Concave-Lens Cases
A concave lens is a diverging lens. For the standard real-object situations considered here, it makes the rays emerging from the lens spread farther apart. In the diagrams below, blue lines are incident rays, red lines are refracted rays, the orange arrow is the object and the purple arrow or point is the image. Dashed purple lines are backward extensions used to locate a virtual image; they do not represent actual light travelling backward. F₁ is the principal focus on the object's side, F₂ is the principal focus on the opposite side and O is the optical centre.
Object at Infinity
An object at a very large distance sends rays that reach the concave lens almost parallel to the principal axis. After refraction, the rays diverge on the opposite side of the lens. Each refracted ray travels as though it came from the first principal focus, F₁, on the object's side. Extend the refracted rays backward using dashed lines; their extensions meet at F₁. Because the actual refracted rays never meet, the image is virtual and cannot be obtained on a screen. In the limiting case of an infinitely distant object, the image is highly diminished and is represented by a point at F₁. It is described as erect because it is not inverted relative to the object.
• Object position: At infinity. • Image position: At F₁ on the object's side of the lens. • Relative size: Highly diminished or point-sized. • Orientation: Erect. • Nature: Virtual. • Screen test: The image cannot be obtained on a screen.
Object at Any Finite Distance
Place the object at any finite distance in front of the concave lens. From the top of the object, draw one ray parallel to the principal axis. After refraction, it diverges and appears to come from F₁. Draw a second ray directed toward F₂ on the opposite side of the lens; after refraction, it travels parallel to the principal axis. The actual refracted rays move apart and cannot form a real image. Extend them backward on the object's side. Their extensions meet between F₁ and O, locating a virtual image. The image arrow is above the principal axis, so it is erect, and shorter than the object arrow, so it is diminished. Since actual light rays do not pass through the image position, the image cannot be obtained on a screen.
• Object position: At any finite distance in front of the lens. • Image position: Between F₁ and O on the object's side. • Relative size: Diminished. • Orientation: Erect. • Nature: Virtual. • Screen test: The image cannot be obtained on a screen. • Motion: Moving the object away moves the image toward F₁ and makes it smaller. Moving the object toward the lens moves the image toward O and makes it larger, although it remains diminished.
Activity
Estimate the convex lens focal length and mark 2F₁, F₁, O, F₂ and 2F₂ on a table. Place a candle at each object position and find the sharp screen image where possible. Then repeat qualitatively with a concave lens and observe directly through it when no screen image forms. Record the complete tables rather than isolated observations.
Quiz
Where does a convex lens form a same-sized real image?
What image forms when a convex-lens object is within F₁?
Which always describes a concave-lens image for a real object?
Why is no finite screen image found for a convex-lens object at F₁?
An enlarged real image beyond 2F₂ requires the object where?
Practice Problems
- Describe a convex-lens image for an object beyond 2F₁. Solution: Between F₂ and 2F₂, diminished, real and inverted.
- Where is an object if its image is at 2F₂ and same-sized? Solution: At 2F₁.
- Why can a magnifier image not be caught on a screen? Solution: With the object inside F₁, refracted rays diverge and only backward extensions meet, so the image is virtual.
- Predict the change as a convex-lens object moves from 2F₁ toward F₁. Solution: The real inverted image moves beyond 2F₂, grows, and tends toward infinity.
- A concave-lens object moves away. Describe the image. Solution: It moves toward F₁ and becomes smaller, remaining virtual and erect between F₁ and O.
Key Takeaways
• A convex lens forms real inverted images for objects outside F₁. • At 2F₁, its real image is same-sized at 2F₂. • Inside F₁, a convex lens forms a virtual erect enlarged image. • At F₁, emergent rays are parallel and the image is at infinity. • A concave lens always forms a virtual erect diminished image for a real object. • Screen capture distinguishes real images from virtual images.