The position, relative size and nature of the image formed by a concave lens for an object placed at infinity are respectively

1. Basics of Refraction and Refractive Index (basic)

Hello! Let’s start our journey into Geometrical Optics by looking at a phenomenon you see every day: a straw appearing bent in a glass of water. This happens because of Refraction. Simply put, refraction is the change in the direction of light as it passes obliquely from one transparent medium (like air) into another (like glass or water). This bending occurs because light travels at different speeds in different materials Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.147.

Refraction follows two fundamental laws. First, the incident ray, the refracted ray, and the 'normal' (an imaginary line perpendicular to the surface) all lie in the same plane. Second, we have Snell’s Law: the ratio of the sine of the angle of incidence (i) to the sine of the angle of refraction (r) is a constant for a given pair of media. This constant is what we call the Refractive Index (n). Mathematically, it is expressed as: n = sin i / sin r Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148.

Crucially, the refractive index is linked to Optical Density. A medium with a higher refractive index is considered 'optically denser,' while one with a lower index is 'optically rarer.' Note that optical density is not the same as mass density! When light travels into a denser medium, it slows down and bends towards the normal. Conversely, when it enters a rarer medium, it speeds up and bends away from the normal Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.149.

Transition Type	Speed Change	Bending Direction
Rarer to Denser (e.g., Air to Glass)	Decreases	Towards the Normal
Denser to Rarer (e.g., Water to Air)	Increases	Away from the Normal

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.147-149

2. Spherical Mirrors: The Foundation of Curved Optics (basic)

Imagine taking a hollow glass sphere and cutting out a small slice. If you polish one of the surfaces to make it reflective, you have created a spherical mirror. Unlike a flat plane mirror, these curved surfaces can manipulate light in fascinating ways, either focusing it to a single point or spreading it out to provide a wide field of view. The most common way to distinguish them is by looking at which side is reflecting: if the reflecting surface curves inwards (like the inside of a spoon), it is a concave mirror; if it bulges outwards (like the back of a spoon), it is a convex mirror Science, Class VIII, p.156.

To master these mirrors, we must first understand their "anatomy." Every spherical mirror has a pole (P), which is the geometric center of the reflecting surface. The mirror is part of a larger imaginary sphere, and the center of that sphere is called the center of curvature (C). The distance between the pole and this center is the radius of curvature (R). An imaginary straight line passing through P and C is the principal axis. When light rays parallel to this axis strike the mirror, they either converge at or appear to diverge from a specific point called the principal focus (F) Science, Class X, Light – Reflection and Refraction, p.136. Interestingly, for mirrors of small aperture, the focal length (f) is exactly half of the radius of curvature (R = 2f).

Feature	Concave Mirror	Convex Mirror
Nature	Converging (brings rays together)	Diverging (spreads rays apart)
Focus	Real (in front of the mirror)	Virtual (behind the mirror)
Primary Use	Shaving mirrors, solar furnaces	Rear-view mirrors in vehicles

To mathematically predict where an image will form, we use the mirror formula: 1/f = 1/v + 1/u. Here, u is the object distance, v is the image distance, and f is the focal length. However, to get the right answer, we must follow the New Cartesian Sign Convention, where distances measured in the direction of incident light are positive, and those against it are negative Science, Class X, Light – Reflection and Refraction, p.143. While a plane mirror always gives an image of the same size, spherical mirrors allow us to magnify or diminish images based on where we place the object Science, Class VIII, p.156.

Sources: Science, Class VIII, Light: Mirrors and Lenses, p.156; Science, Class X, Light – Reflection and Refraction, p.136; Science, Class X, Light – Reflection and Refraction, p.143

3. Introduction to Spherical Lenses (basic)

At its simplest, a spherical lens is a piece of transparent material (usually glass or plastic) bound by two surfaces, where at least one surface is spherical. Depending on their shape, lenses are categorized into two primary types: convex and concave. A convex lens is thicker at the middle than at the edges and is known as a converging lens because it brings parallel rays of light together at a single point. Conversely, a concave lens is thinner at the middle and thicker at the edges; it is called a diverging lens because it spreads out parallel light rays as they pass through Science, Class VIII, Light: Mirrors and Lenses, p.164.

To understand how these lenses work, we must recognize a few key anatomical features. Every lens has an optical centre (O), which is its central point. A unique property of the optical centre is that any ray of light passing through it emerges without any deviation from its original path Science, class X, Light – Reflection and Refraction, p.154. Furthermore, because a lens has two surfaces, it has two principal foci (represented as F₁ and F₂), one on either side. The distance between the optical centre and the principal focus is known as the focal length (f) Science, class X, Light – Reflection and Refraction, p.151.

The effectiveness of a lens is measured by its Power (P), which is defined as the reciprocal of its focal length (P = 1/f). A lens with a very short focal length is "stronger" because it bends light rays more sharply. For instance, a convex lens with a short focal length focuses light closer to the optical centre compared to one with a long focal length Science, class X, Light – Reflection and Refraction, p.157. This relationship is crucial in designing everything from corrective eyeglasses to advanced telescopes.

Feature	Convex Lens	Concave Lens
Shape	Bulging outwards (thick center)	Curved inwards (thin center)
Effect on Light	Converging	Diverging
Common Name	Converging Lens	Diverging Lens

Sources: Science, Class VIII, Light: Mirrors and Lenses, p.164; Science, class X, Light – Reflection and Refraction, p.151, 154, 157

4. Defects of Human Vision and Correction (intermediate)

To understand vision defects, we must first appreciate the Power of Accommodation—the eye's ability to adjust its focal length using ciliary muscles. When the eye's refractive system (the cornea and lens) fails to focus light exactly on the retina, the resulting image is blurred. The most common refractive defects are Myopia, Hypermetropia, and Presbyopia. Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.163

Myopia (Nearsightedness) occurs when a person can see nearby objects clearly but distant objects appear blurred. This happens because the image is formed in front of the retina, often due to excessive curvature of the eye lens or elongation of the eyeball. To correct this, we use a concave (diverging) lens of suitable power, which diverges the incoming parallel rays just enough so that, after passing through the eye lens, they focus precisely on the retina. Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.163

Hypermetropia (Farsightedness) is the opposite: distant objects are clear, but nearby objects are blurry. Here, the light rays focus behind the retina, typically because the eyeball is too short or the focal length of the eye lens is too long. This is corrected using a convex (converging) lens, which provides the additional refractive power needed to bring the image forward onto the retina. Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.163

Defect	Image Formation	Corrective Lens	Lens Nature
Myopia	In front of retina	Concave	Diverging (Negative Power)
Hypermetropia	Behind retina	Convex	Converging (Positive Power)

As we age, we encounter Presbyopia. This arises because the ciliary muscles weaken and the eye lens loses its flexibility, causing the near point to recede. Many seniors suffer from both myopia and hypermetropia simultaneously. Such cases require bi-focal lenses, where the upper portion is concave (for distance) and the lower portion is convex (for reading/near vision). Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.164

Sources: Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.163; Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.164

5. Lens Formula and Power of Lenses (intermediate)

When light passes through a lens, its path is altered based on the lens's curvature and material. To quantify this precisely, we use the Lens Formula, which establishes a mathematical relationship between the object distance (u), the image distance (v), and the focal length (f). The formula is expressed as: 1/v - 1/u = 1/f. It is important to remember that this formula is universal for all spherical lenses, whether they are thin convex or thin concave lenses Science, Chapter 9, p.155. However, to get the correct answer in numerical problems, you must strictly follow the New Cartesian Sign Convention: distances measured in the direction of incident light (usually to the right of the lens) are positive, while those against it are negative.

The concept of Power of a Lens relates to its ability to converge or diverge light rays. A lens with a short focal length bends light more sharply, meaning it has greater "power." Mathematically, power (P) is the reciprocal of the focal length (f) expressed in meters (P = 1/f). The SI unit of power is the dioptre (D), where 1 D = 1 m⁻¹ Science, Chapter 9, p.158. For instance, a lens with a focal length of 2 meters has a power of 0.5 D.

Lens Type	Focal Length (f)	Power (P)	Nature
Convex	Positive (+)	Positive (+)	Converging
Concave	Negative (-)	Negative (-)	Diverging

In practical optics, such as eye examinations, doctors often prescribe a combination of lenses. The net power (P) of a system of lenses placed in contact is simply the algebraic sum of the individual powers: P = P₁ + P₂ + P₃... This additive property allows opticians to design complex lens systems by simply adding or subtracting thin lenses to reach the desired corrective strength Science, Chapter 10, p.170.

Sources: Science, Chapter 9: Light – Reflection and Refraction, p.155, 158, 159; Science, Chapter 10: The Human Eye and the Colourful World, p.170

6. Image Formation by Concave Lenses (exam-level)

A concave lens, often referred to as a diverging lens, is thinner at the center than at the edges. Its primary characteristic is that it spreads out (diverges) light rays that travel parallel to its principal axis. To understand how it forms images, we must first look at its geometry: it consists of two spherical surfaces with centers of curvature (C₁ and C₂), and an imaginary line passing through these centers called the principal axis Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.150. When parallel rays of light strike a concave lens, they refract and move away from each other. However, if we trace these refracted rays backward, they appear to meet at a single point on the same side as the incoming light, known as the principal focus (F₁).

The image formation process in a concave lens is remarkably consistent compared to its convex counterpart. Regardless of where you place the object, the lens always produces an image that is virtual, erect, and diminished. A virtual image means the light rays do not actually intersect; they only appear to do so when projected backward. Because these rays never truly meet on the other side of the lens, a concave lens can never form a real image on a screen Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.154.

There are two primary scenarios to master for your exams:

• Object at Infinity: The incident rays are parallel to the principal axis. After refraction, they diverge and appear to meet at the focus (F₁). The resulting image is point-sized (highly diminished), virtual, and erect Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.156.
• Object between Infinity and Optical Center: As the object moves closer, the image remains virtual and erect but forms between the focus (F₁) and the optical center (O). The size remains smaller than the object itself.

Object Position	Image Position	Relative Size	Nature of Image
At infinity	At focus F₁	Highly diminished (point-sized)	Virtual and erect
Between infinity and O	Between F₁ and O	Diminished	Virtual and erect

Sources: Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.150; Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.153; Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.154; Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.156

7. Solving the Original PYQ (exam-level)

Now that you've mastered the principles of ray optics, this question tests your ability to apply the diverging nature of a concave lens. When an object is at infinity, the incident rays are parallel to the principal axis. According to the rules of refraction you just studied, these parallel rays diverge upon passing through the lens. To an observer, they appear to originate from a single point—the principal focus (F1). Since the light rays do not actually intersect but only appear to do so, the image formed is virtual and erect. This fundamental behavior ensures the image is highly diminished (point-sized) specifically at the focus.

To arrive at (A) at focus, diminished and virtual, you should use the process of elimination—a vital UPSC strategy. Options (B) and (D) can be immediately discarded because they mention real images; remember, a concave lens always produces virtual images because the rays diverge. The trap in option (C) is the position; while concave lenses do form images between the focus and optical center for objects at finite distances, the specific condition of "infinity" always dictates that the rays must relate to the focus. As detailed in Science, Class X (NCERT 2025 ed.), understanding these geometric limits is key to avoiding such distractors.