In a convex lens, when the object is at infinity, then the image is formed at ________.
Let's understand how image formation works with a convex lens, especially when the object is placed very far away, effectively at infinity.
Understanding Image Formation by a Convex Lens
A convex lens, also known as a converging lens, is thicker in the middle than at the edges. It converges parallel rays of light that fall on it to a point. This converging property is key to how it forms images.
Key terms related to a convex lens include:
- Principal Axis: The straight line passing through the optical centre and perpendicular to both faces of the lens.
- Optical Centre (O): The central point of the lens. A ray of light passing through the optical centre goes undeviated.
- Principal Focus (F₁, F₂): There are two principal foci, one on each side of the lens. F₁ is on the object side, and F₂ is on the image side. Parallel rays incident on the lens converge at F₂ after refraction. Rays passing through F₁ become parallel to the principal axis after refraction.
- Focal Length (f): The distance between the optical centre and the principal focus.
- 2F₁ and 2F₂: Points at twice the focal length on either side of the lens. These points are sometimes referred to as the centres of curvature if the lens surfaces were part of spheres.
Image Formation when Object is at Infinity
When an object is placed at infinity, it means the object is so far away that the light rays coming from it and incident on the lens are practically parallel to each other. For objects at infinity, these parallel rays are generally considered parallel to the principal axis for simplicity in diagrams, although they can also be parallel but inclined to the principal axis.
Consider the case where parallel rays from an object at infinity are incident parallel to the principal axis:
- All rays incident on the convex lens parallel to the principal axis, after refraction through the lens, converge at the principal focus on the other side of the lens. This point is denoted as F₂.
- Therefore, the image of an object at infinity is formed at the principal focus F₂.
The image formed in this case has specific characteristics:
- Position: At the principal focus F₂ on the opposite side of the lens.
- Nature: Real and inverted. Real because the rays actually converge at this point. Inverted relative to the object (though for an object at infinity, size is point-like).
- Size: Highly diminished, effectively a point image.
Analysing the Given Options
The question asks for the location of the image when the object is at infinity for a convex lens. Based on the principles of image formation:
- F 2: This represents the principal focus on the image side. As discussed, parallel rays from infinity converge at the principal focus. This aligns with our understanding.
- Beyond 2F 2 : This region is further from the lens than twice the focal length on the image side. Images are formed here for objects placed between F₁ and 2F₁.
- At infinity: This would imply the rays become parallel after passing through the lens. This happens when the object is placed at the principal focus F₁ on the object side.
- Between F2 and 2F 2 : This region is between the principal focus and twice the focal length on the image side. Images are formed here for objects placed beyond 2F₁ on the object side.
The correct location for the image of an object at infinity formed by a convex lens is indeed at the principal focus F₂.
Comparing Image Formation Positions
The position of the object relative to the convex lens determines the position, nature, and size of the image. Here's a quick look at how the image position changes:
| Object Position | Image Position | Image Nature | Image Size |
|---|---|---|---|
| At infinity | At F₂ | Real and Inverted | Highly Diminished (point) |
| Beyond 2F₁ | Between F₂ and 2F₂ | Real and Inverted | Diminished |
| At 2F₁ | At 2F₂ | Real and Inverted | Same size |
| Between F₁ and 2F₁ | Beyond 2F₂ | Real and Inverted | Magnified |
| At F₁ | At infinity | Real and Inverted (at infinity) | Highly Magnified |
| Between O and F₁ | On the same side as object | Virtual and Erect | Magnified |
This table clearly shows that when the object is at infinity, the image is formed at F₂.
Revision Table: Convex Lens Image Formation Summary
| Object Location | Image Location | Nature of Image | Size of Image |
|---|---|---|---|
| Infinity ($\infty$) | At F₂ | Real, Inverted | Point size (Highly diminished) |
| Beyond 2F₁ | Between F₂ and 2F₂ | Real, Inverted | Diminished |
| At 2F₁ | At 2F₂ | Real, Inverted | Same size |
| Between F₁ and 2F₁ | Beyond 2F₂ | Real, Inverted | Magnified |
| At F₁ | Infinity ($\infty$) | Real, Inverted | Highly Magnified |
| Between F₁ and Optical Centre (O) | On the same side as object | Virtual, Erect | Magnified |
Additional Information: Key Concepts in Lens Optics
Understanding a few more terms can help solidify the concepts of lens optics:
- Focal Plane: A plane passing through the principal focus and perpendicular to the principal axis. When parallel rays are incident on the lens but are inclined to the principal axis, they converge at a point in the focal plane.
- Ray Diagrams: These diagrams use specific rays to determine the position, size, and nature of the image formed by a lens. Common rays used are: a ray parallel to the principal axis passes through F₂ after refraction; a ray passing through F₁ becomes parallel to the principal axis after refraction; a ray passing through the optical centre goes undeviated.
- Lens Formula: The mathematical relationship between object distance ($\text{u}$), image distance ($\text{v}$), and focal length ($\text{f}$) is given by the lens formula: $\frac{1}{\text{v}} - \frac{1}{\text{u}} = \frac{1}{\text{f}}$. For an object at infinity, $\text{u} = -\infty$ (by convention, object distance is negative). Substituting this into the formula, $\frac{1}{\text{v}} - \frac{1}{-\infty} = \frac{1}{\text{f}}$, which simplifies to $\frac{1}{\text{v}} - 0 = \frac{1}{\text{f}}$, so $\text{v} = \text{f}$. This confirms that the image is formed at the focal length distance from the lens, which is the location of F₂ (on the image side).
The case of an object at infinity and the image forming at the principal focus is a fundamental concept in lens optics and is used in applications like telescopes.
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