If the focal length of the biconvex lens is 25 cm, then the power of the lens will be

1. Nature of Light and Refraction (basic)

Welcome to your first step in mastering Geometrical Optics! To understand how lenses and mirrors work, we must first understand the nature of light. In our everyday experience, light appears to travel in straight lines—a concept known as rectilinear propagation Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.134. However, the most fascinating behavior occurs when light transitions from one transparent medium (like air) to another (like glass or water).

This phenomenon is called Refraction. Refraction is the bending of light as it passes obliquely from one transparent medium to another. Why does it bend? It happens because light changes its speed. While light travels at its maximum speed of approximately 3 × 10⁸ m/s in a vacuum, it slows down significantly in denser materials like glass or water Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148. This change in speed causes the ray to pivot at the boundary, changing its direction.

The "strength" of this bending is governed by the Refractive Index (n). It is a unitless constant that represents the ratio of the speed of light in the two media. Specifically, the refractive index of medium 2 with respect to medium 1 is the ratio of the speed of light in medium 1 (v₁) to the speed of light in medium 2 (v₂). As per Snell’s Law, this is also expressed as the ratio of the sine of the angle of incidence (i) to the sine of the angle of refraction (r) Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148.

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.134; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.147; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148

2. Spherical Lenses: Types and Properties (basic)

In our journey through optics, we now move from mirrors to spherical lenses. A lens is a piece of transparent material (like glass or plastic) bound by two surfaces, where at least one surface is spherical. Unlike mirrors which reflect light, lenses refract (bend) light as it passes through them Science, Class X (NCERT 2025 ed.), Chapter 9, p.150.

There are two primary types of spherical lenses you must master for the UPSC syllabus: Convex and Concave. A Convex lens is thicker at the middle than at the edges; it converges parallel rays of light to a single point, which is why it is often called a converging lens. Conversely, a Concave lens is thinner in the middle and thicker at the edges. It causes parallel rays to spread out, earning it the name diverging lens Science, Class VIII (NCERT 2025 ed.), Light: Mirrors and Lenses, p.164.

To quantify how strongly a lens can bend light, we use a property called Power (P). Power is defined as the reciprocal of the focal length (f) when measured in meters: P = 1/f (in m). The SI unit of power is the dioptre (D). Following the sign convention, a convex lens has a positive power because its focal length is positive, while a concave lens has a negative power Science, Class X (NCERT 2025 ed.), Chapter 9, p.157. For instance, if you have a convex lens with a focal length of 25 cm (which is 0.25 m), its power would be +4 D (1 ÷ 0.25 = 4).

Sources: Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.150, 157; Science, Class VIII (NCERT 2025 ed.), Light: Mirrors and Lenses, p.164

3. The Cartesian Sign Convention (intermediate)

In geometrical optics, we don't just care about how far an object is; we care about which side of the lens or mirror it sits on. To make our mathematical formulas (like the lens or mirror formulas) work universally, we use the New Cartesian Sign Convention. This system treats the optical system like a graph. For mirrors, the pole (P) is the origin; for lenses, the optical centre (O) is the origin Science, Light – Reflection and Refraction, p.142. The principal axis acts as the x-axis (X'X) of our coordinate system.

The most fundamental rule is that the object is always placed to the left of the mirror or lens. This means incident light always travels from left to right. From this, we derive the rules for signs:

• Distances measured in the direction of incident light (to the right of the origin) are taken as positive.
• Distances measured against the direction of incident light (to the left of the origin) are taken as negative.
• Heights measured upward and perpendicular to the principal axis are positive.
• Heights measured downward and perpendicular to the principal axis are negative Science, Light – Reflection and Refraction, p.143.

This convention explains why certain values are fixed in our calculations. For instance, because the object is always on the left, the object distance (u) is always negative. For lenses, the convention specifically dictates that the focal length of a convex lens is positive, while the focal length of a concave lens is negative Science, Light – Reflection and Refraction, p.155. Mastering these signs is the difference between a correct derivation and a complete numerical disaster in physics!

Sources: Science, Light – Reflection and Refraction, p.142; Science, Light – Reflection and Refraction, p.143; Science, Light – Reflection and Refraction, p.155

4. The Human Eye and Vision Defects (intermediate)

The human eye is essentially a sophisticated biological camera that uses a living lens system to form images on a light-sensitive screen called the retina Science, The Human Eye and the Colourful World, p.161. Light enters through a transparent membrane called the cornea, and the iris (the colored part) acts as a shutter, adjusting the pupil size to control light intensity. Unlike a glass camera lens that moves back and forth to focus, our eye lens is flexible. It changes its curvature—and thus its focal length—through the action of ciliary muscles. This remarkable ability is known as the Power of Accommodation Science, The Human Eye and the Colourful World, p.162.

For a young adult with normal vision, the near point (the closest distance for clear vision without strain) is approximately 25 cm, while the far point is at infinity Science, The Human Eye and the Colourful World, p.162. When the ciliary muscles are relaxed, the lens becomes thin, increasing its focal length to see distant objects. Conversely, when looking at nearby objects, the muscles contract, making the lens thicker and decreasing the focal length to ensure the image still lands precisely on the retina.

Vision defects occur when the eye's refractive system fails to focus light exactly on the retina. The two most common refractive errors are Myopia and Hypermetropia. In Myopia, the image is formed in front of the retina, often because the eyeball is too long. In Hypermetropia, the image is formed behind the retina, often because the eyeball is too short or the lens cannot become thick enough Science, The Human Eye and the Colourful World, p.163.

Sources: Science, The Human Eye and the Colourful World, p.161; Science, The Human Eye and the Colourful World, p.162; Science, The Human Eye and the Colourful World, p.163

5. Corrective Lenses and Applications (intermediate)

To understand corrective lenses, we must first master the concept of the Power of a Lens (P). Power is a measure of the degree of convergence or divergence a lens can achieve. Scientifically, it is defined as the reciprocal of its focal length (f) when measured in meters: P = 1/f. The SI unit for power is the dioptre (D). When a lens has a focal length of 1 meter, its power is 1 D. Crucially, the sign of the power tells us the nature of the lens: convex (converging) lenses have positive power, while concave (diverging) lenses have negative power Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.157.

In medical applications, these lenses are used to correct refractive defects by shifting where light focuses within the eye. In Myopia (near-sightedness), the image forms in front of the retina; a concave lens is used to diverge the incoming light so that the focus is pushed back onto the retina Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.163. Conversely, in Hypermetropia (far-sightedness), the image forms behind the retina. A convex lens provides the "additional focusing power" required to pull the image forward onto the retina Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.164.

For more complex cases like Presbyopia, where a person struggles with both near and distant vision due to the weakening of ciliary muscles, bi-focal lenses are used. These specialized lenses contain both types: the upper portion is concave for distance, and the lower portion is convex for reading Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.164.

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.157; Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.163-164

6. Lens Formula and Power of a Lens (exam-level)

When we work with spherical lenses, we need a reliable way to predict where an image will form based on where an object is placed. This is where the Lens Formula comes in. It provides a mathematical relationship between the object-distance (u), the image-distance (v), and the focal length (f) of the lens. The formula is expressed as:

1/v - 1/u = 1/f

It is vital to remember that this formula is universal—it applies to both convex and concave lenses in all situations. However, its accuracy depends entirely on the consistent use of the New Cartesian Sign Convention. For instance, the object distance (u) is almost always negative because the object is placed to the left of the lens. Science, Chapter 9: Light – Reflection and Refraction, p.155

Moving beyond just finding the image, we often need to measure a lens's ability to bend light. This is known as the Power of a Lens (P). Simply put, power is the degree of convergence or divergence a lens produces. A lens with a short focal length bends light rays more sharply, meaning it is more "powerful." Mathematically, power is the reciprocal of the focal length (measured in meters):

P = 1/f (in meters)

The SI unit of power is the dioptre, denoted by the letter D. One dioptre (1 D) is defined as the power of a lens whose focal length is exactly 1 meter. Science, Chapter 9: Light – Reflection and Refraction, p.157-158

In clinical practice and exams, the sign of the power tells you the nature of the lens immediately. A convex lens, which converges light, has a positive focal length and thus a positive power. Conversely, a concave lens, which diverges light, has a negative focal length and a negative power. If an optician prescribes a +2.0 D lens, you know instantly it is a convex lens used for correction. Science, Chapter 9: Light – Reflection and Refraction, p.158

Sources: Science, Chapter 9: Light – Reflection and Refraction, p.155; Science, Chapter 9: Light – Reflection and Refraction, p.157; Science, Chapter 9: Light – Reflection and Refraction, p.158

7. Solving the Original PYQ (exam-level)

Now that you have mastered the building blocks of optics—specifically lens geometry and sign conventions—this question allows you to apply those concepts in a practical scenario. A biconvex lens is structurally a convex lens, which we know acts as a converging lens. According to the Cartesian sign convention detailed in Science, class X (NCERT 2025 ed.), the focal length of a converging lens is always positive. Identifying this 'nature' of the lens is your first strategic step, immediately narrowing your viable options to the positive values before you even perform a calculation.

To find the numerical value, we use the Power of a lens (P) formula: P = 1/f. The most critical trap here is the unit of measurement; power is measured in dioptres (D) only when the focal length is expressed in meters. Converting the given 25 cm to meters gives us 0.25 m (25/100). When we divide 1 by 0.25, we get 4. By combining our numerical result with our established positive sign, we arrive at (A) + 4 dioptre as the correct answer.

UPSC often uses distractors to catch common procedural errors. Option (B) - 4 dioptre is a sign trap, designed for students who forget that biconvex lenses are converging (positive). Options (C) and (D) are unit conversion traps; these occur if a candidate fails to convert centimeters to meters and simply calculates 1/25, resulting in 0.04. In the exam, always perform a quick 'sanity check': a focal length of 25 cm indicates a relatively strong lens, so a power of 4 is much more logical than a near-zero power of 0.04.