Which one of the following four glass lenses is a diverging lens?

1. Fundamentals of Refraction and Light (basic)

Welcome to our journey into Geometrical Optics! To understand how lenses work, we must first master the root phenomenon: Refraction. In simple terms, refraction is the bending of light when it travels from one transparent medium to another. Think of a car moving from a smooth highway onto a sandy patch at an angle; the wheels hitting the sand first will slow down, causing the car to swerve. Similarly, light changes its speed when it enters a different material, which causes it to change direction Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.134.

The behavior of this bending is governed by the Laws of Refraction. The most famous of these is Snell’s Law, which states that 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 known as the Refractive Index (n). Mathematically, it is expressed as:
sin i / sin r = constant (n) Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148.

It is crucial to distinguish between "mass density" and "optical density." A material might be lightweight but still slow light down significantly. For instance, kerosene has a higher refractive index (1.44) than water (1.33), meaning it is optically denser even though it floats on water Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.149. When light enters an optically denser medium, it slows down and bends towards the normal; when it enters a rarer medium, it speeds up and bends away from the normal.

Medium Type	Speed of Light	Refractive Index (n)	Bending Direction
Optically Rarer (e.g., Air)	Higher	Lower (~1.00)	Away from Normal
Optically Denser (e.g., Glass)	Lower	Higher (~1.50)	Towards Normal

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.134; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.149

2. Spherical Mirrors: Converging vs. Diverging Reflection (basic)

To understand spherical mirrors, imagine a hollow glass sphere. If you cut a piece out of it and silver one side, you create a spherical mirror. The nature of how light reflects off these surfaces depends entirely on which side is polished. As we learn in Science, Class VIII, Light: Mirrors and Lenses, p.155, we can identify these mirrors by looking at them from the side: a concave mirror has its reflecting surface curved inwards (like the inside of a spoon), while a convex mirror has its reflecting surface bulging outwards.

The defining characteristic of these mirrors is how they treat parallel rays of light. A concave mirror is known as a converging mirror because when parallel rays of light hit its surface, they reflect and meet (converge) at a single point called the principal focus. Conversely, a convex mirror is known as a diverging mirror. When parallel rays hit a convex surface, they reflect and spread out (diverge) in different directions. If you were to trace these reflected rays backward, they would only appear to meet at a point behind the mirror (Science, Class VIII, Light: Mirrors and Lenses, p.165).

Feature	Concave Mirror	Convex Mirror
Shape	Reflecting surface curved inwards.	Reflecting surface curved outwards.
Action on Light	Converging (rays meet).	Diverging (rays spread out).
Common Use	Headlights, shaving mirrors.	Rear-view mirrors in vehicles.

Crucially, the laws of reflection—where the angle of incidence equals the angle of reflection—apply to every single point on these curved surfaces just as they do on a flat plane mirror (Science, Class VIII, Light: Mirrors and Lenses, p.165). The curvature simply changes the direction of the 'normal' line at every point, which is what causes the rays to either bundle together or scatter away.

Sources: Science, Class VIII, NCERT (Revised ed 2025), Light: Mirrors and Lenses, p.155; Science, Class VIII, NCERT (Revised ed 2025), Light: Mirrors and Lenses, p.165

3. Basic Terminology of Spherical Lenses (basic)

To understand spherical lenses, we must first look at their physical geometry. A lens is a transparent material bound by two surfaces, at least one of which is spherical. The way these surfaces curve determines how the lens interacts with light. The most fundamental way to classify a lens is by comparing its central thickness to its edge thickness. A lens that is thicker at the middle than at the edges is a convex lens (converging), while one that is thinner at the middle than at the edges is a concave lens (diverging).

Let’s break down the essential terminology you need to master:

• Optical Centre (O): This is the exact central point of the lens. A unique property of the optical centre is that any ray of light passing through it emerges without any deviation from its path Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.150, 154.
• Principal Focus (F) and Focal Length (f): Because a lens has two surfaces, it has two principal foci, represented as F₁ and F₂. For a convex lens, parallel rays actually meet at the focus; for a concave lens, they only appear to diverge from it Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.151, 154. The distance from the optical centre to the principal focus is the focal length.
• Power of a Lens (P): This measures the degree of convergence or divergence. It is the reciprocal of the focal length (P = 1/f) and is measured in Dioptres (D) Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158.

The following table helps distinguish between the two primary types of spherical lenses:

Feature	Convex Lens	Concave Lens
Shape	Thicker at the center	Thinner at the center
Nature	Converging (brings rays together)	Diverging (spreads rays out)
Power/Focal Length Sign	Positive (+)	Negative (–)

Sources: Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.150; Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.151; Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.154; Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158

4. Vision Defects and Corrective Lenses (intermediate)

In the world of optics, our eyes are sophisticated cameras that use a flexible lens to focus light onto the retina. The ability of the eye to adjust its focal length to see both nearby and distant objects is known as the power of accommodation Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.170. However, when the eye lens cannot properly focus light—either due to the shape of the eyeball or the aging of ciliary muscles—vision becomes blurred. These are known as refractive defects.

There are three primary defects you must master for your exams: Myopia, Hypermetropia, and Presbyopia.

• Myopia (Near-sightedness): A person can see nearby objects clearly but struggles with distant ones. This happens because the light rays from a distance converge in front of the retina rather than on it. To fix this, we use a concave lens (a diverging lens) which spreads the rays out before they enter the eye, allowing them to focus exactly on the retina Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.170.
• Hypermetropia (Far-sightedness): Distant objects are clear, but nearby objects are blurry. The image is formed behind the retina because the eyeball is too short or the lens focal length is too long Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.163. This is corrected using a convex lens (a converging lens).
• Presbyopia: Often called 'old-age far-sightedness,' this occurs when the eye loses its power of accommodation with age, making it difficult to see nearby objects Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.163.

To identify these lenses physically, remember their geometry. A converging (convex) lens is thicker at the center than at the edges. Conversely, a diverging (concave) lens is thinner at the center compared to its periphery Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.150. This physical shape determines whether the light rays will be brought together or spread apart.

Defect	Focus Point	Corrective Lens	Lens Shape
Myopia	In front of Retina	Concave (Diverging)	Thinner at center
Hypermetropia	Behind Retina	Convex (Converging)	Thicker at center

Sources: Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.162, 163, 170; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.150

5. Optical Instruments in Science & Technology (intermediate)

To understand optical instruments, we must first look at how lenses manipulate light. A lens's behavior—whether it makes light rays meet at a point or spread apart—is governed entirely by its geometry. A converging lens (convex) is thicker at the center than at the edges, which causes parallel rays to bend inward. Conversely, a diverging lens (concave) is thinner at the center than at its periphery, forcing parallel rays to refract away from the optical axis. This fundamental principle allows us to design everything from simple eyeglasses to complex smartphone cameras Science, Class VIII NCERT, Light: Mirrors and Lenses, p.165.

Feature	Converging Lens (Convex)	Diverging Lens (Concave)
Physical Shape	Thicker in the middle	Thinner in the middle
Action on Light	Converges rays to a focal point	Spreads rays outward (diverges)
Common Use	Magnifying glasses, human eye lens	Correcting nearsightedness

In sophisticated optical instruments like microscopes and telescopes, a single lens is rarely enough. Engineers use a combination of lenses placed in contact to minimize image defects (aberrations) and achieve the desired magnification. The total Optical Power (P) of such a system is simply the algebraic sum of the individual powers: P = P₁ + P₂ + ... Science, Class X NCERT, Light – Reflection and Refraction, p.158. This additive property is crucial; for instance, combining a lens of +2.0 D and -0.5 D results in a net power of +1.5 D, allowing for precise control over how the instrument focuses light. While lenses are the stars of refraction, many modern reflecting telescopes actually rely on curved mirrors. A large concave mirror is often used as the primary objective to gather light from distant stars, as mirrors can be made much larger and lighter than glass lenses Science, Class VIII NCERT, Light: Mirrors and Lenses, p.156. Whether using lenses or mirrors, the goal remains the same: to capture and redirect light to create a clear, useful image for the human eye.

Sources: Science, Class VIII NCERT, Light: Mirrors and Lenses, p.156, 165; Science, Class X NCERT, Light – Reflection and Refraction, p.158

6. Classification of Lenses by Geometry (intermediate)

When we classify lenses by their geometry, we are essentially looking at how their physical thickness varies from the center to the periphery. A lens is a transparent material bound by two surfaces, where at least one surface is spherical Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.150. The most fundamental rule to remember is this: the behavior of a lens (whether it converges or diverges light) is determined by whether it is thicker or thinner in the middle compared to its edges.

Lenses are broadly categorized into two families based on this geometric relationship:

• Convex (Converging) Lenses: These are thicker at the middle than at the edges. When parallel rays of light pass through them, they are refracted inward toward the principal axis, meeting at a point. Common types include biconvex (bulging on both sides), plano-convex (one flat side), and converging meniscus Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.150.
• Concave (Diverging) Lenses: These are thicker at the edges than at the middle. Because the central portion is thinner, parallel rays of light are refracted outward, away from the principal axis, appearing to spread out or "diverge" from a virtual point. A double concave lens is the most common example Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.150.

Lens Type	Geometry (Center vs. Edges)	Optical Effect
Convex	Thicker at the center	Converging (rays meet)
Concave	Thinner at the center	Diverging (rays spread)

It is also important to note the terminology of the lens structure. Every spherical surface of a lens forms part of a sphere, and the centers of these spheres are known as centers of curvature (C₁ and C₂). An imaginary line passing through these centers is the principal axis Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.150. Even complex shapes like a meniscus lens (which has one convex and one concave side) follow the thickness rule: if the center is thicker than the edges, it converges light; if the center is thinner, it diverges light.

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.150

7. Solving the Original PYQ (exam-level)

This question is a perfect application of the fundamental principles of refraction and lens geometry you just mastered. The core concept here is the thickness gradient: the behavior of light (whether it converges or diverges) is determined by whether the lens is thicker or thinner at its center compared to its edges. According to NCERT Class 10 Science, a lens that spreads out parallel rays of light is a diverging lens, and this occurs geometrically when the central portion is thinner than the periphery.

To arrive at the correct answer, look closely at the cross-sections provided. In diagram (ii), the lens curves inward, creating a double concave shape where the middle is visibly narrower than the top and bottom edges. This specific geometry causes light rays to refract away from the optical axis. Therefore, Option (B) ii is the correct choice. As a coach, I advise you to always visualize the ray diagram: if the center is thin, the light "bends away," making it a diverging system.

UPSC often includes "distractor" shapes to test your consistency. Options (i) and (iii) are converging lenses (biconvex and plano-convex) because their centers are thicker, which funnels light inward. The most common trap is Option (iv), the positive meniscus. Even though it is curved, its center remains thicker than its tapered ends, meaning it still acts as a converging lens. Don't be fooled by the crescent shape; always apply the center-vs-edge thickness rule to identify the nature of the lens accurately.