Wh at is the power of the lens, if the far point of a short-sighted eye is 200 cm?

1. Fundamental Laws of Light: Refraction and Spherical Lenses (basic)

Welcome to your first step into the world of Geometrical Optics! To understand how we see the world—and how we correct our vision—we must first master how light behaves when it travels from one material to another. This phenomenon is known as Refraction. Unlike reflection, where light bounces back, refraction occurs because light changes its speed when entering a different medium, causing it to bend Science, Class X, p.134.

Refraction is governed by two fundamental laws. First, the incident ray, the refracted ray, and the "normal" (the perpendicular line at the point of contact) all lie in the same plane. Second, we have 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 Science, Class X, p.148. This constant is known as the Refractive Index (n). It tells us how much a medium can "bend" light; for instance, light bends much more in diamond (n = 2.42) than in water (n = 1.33) Science, Class X, p.149.

Movement of Light	Bending Direction	Speed Change
Rarer to Denser (e.g., Air to Glass)	Towards the Normal	Decreases
Denser to Rarer (e.g., Glass to Air)	Away from the Normal	Increases

When we shape transparent materials into Spherical Lenses, we can control this bending to converge or diverge light. A Convex lens (thicker at the middle) converges light to a point, while a Concave lens (thinner at the middle) spreads it out. The strength of this bending is measured as the Power of a Lens (P), which is simply the reciprocal of its focal length (f) in meters: P = 1/f Science, Class X, p.157. A lens with a short focal length bends light more sharply and thus has higher power.

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; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.157

2. Sign Convention and Lens Formula (intermediate)

In geometrical optics, we need a consistent "mathematical grammar" to describe how light behaves. This is provided by the New Cartesian Sign Convention. Just like in a coordinate geometry graph, we treat the optical centre (O) of the lens as the origin (0,0) and the principal axis as the X-axis. According to this convention, the object is always placed to the left of the lens. This means light travels from left to right. Consequently, any distance measured in the direction of the incident light (to the right of the origin) is taken as positive, while distances measured against the direction of incident light (to the left) are negative Science, Light – Reflection and Refraction, p.142.

The Lens Formula is the core mathematical relationship that links the object-distance (u), the image-distance (v), and the focal length (f). It is expressed as:

1/v – 1/u = 1/f

It is vital to distinguish this from the mirror formula (which uses a plus sign). This formula is universal, applying to both convex and concave lenses in all situations Science, Light – Reflection and Refraction, p.155. When solving problems, you must substitute the values of u, v, and f with their proper signs. For instance, since the object is placed on the left, u is almost always negative in standard problems.

Quantity	Convex Lens (Converging)	Concave Lens (Diverging)
Focal Length (f)	Positive (+)	Negative (–)
Object Distance (u)	Negative (–)	Negative (–)

Finally, we define the Power of a lens (P) as the reciprocal of its focal length in meters (P = 1/f). The SI unit is the Dioptre (D). Because of the sign convention, a convex lens has positive power, while a concave lens has negative power Science, Light – Reflection and Refraction, p.159.

Sources: Science, Light – Reflection and Refraction, p.142; Science, Light – Reflection and Refraction, p.155; Science, Light – Reflection and Refraction, p.159

3. Anatomy of the Human Eye and Power of Accommodation (basic)

The human eye is essentially a biological camera, but with a sophisticated, self-adjusting lens system. When light enters the eye, it first passes through the cornea, a transparent bulge on the front of the eyeball. Interestingly, most of the light's refraction (bending) occurs at this outer surface, rather than in the lens itself Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.161. Behind the cornea lies the iris—the colored part of your eye—which acts like a shutter by controlling the pupil's size to regulate how much light enters.

The crystalline lens is a flexible, jelly-like structure that performs the "fine-tuning" of focus. Unlike a glass camera lens that moves forward or backward to focus, our eye lens changes its curvature to adjust its focal length. This remarkable ability is called the Power of Accommodation. This adjustment is managed by the ciliary muscles. When these muscles are relaxed, the lens stays thin, allowing us to see distant objects. To see something close (like a book), the ciliary muscles contract, making the lens thicker and decreasing its focal length Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.162.

Feature	Viewing Distant Objects	Viewing Nearby Objects
Ciliary Muscles	Relaxed	Contracted
Lens Shape	Thin (less curvature)	Thick (more curvature)
Focal Length	Increases	Decreases

Finally, the lens forms a real and inverted image on the retina, a membrane packed with light-sensitive cells. These cells convert light into electrical signals, which the optic nerve carries to the brain. For a healthy young adult, the closest distance at which an object can be seen clearly without strain—the least distance of distinct vision—is approximately 25 cm Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.170.

Sources: Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.161; Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.162; Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.170

4. Hypermetropia, Presbyopia, and Astigmatism (intermediate)

To understand vision defects, we must first appreciate the eye's power of accommodation — its ability to adjust the focal length of the crystalline lens to see both near and distant objects. When the eye loses this ability or the eyeball's physical dimensions are slightly off, refractive defects occur Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.162. We focus here on the three primary conditions that hinder near-vision or overall clarity.

Hypermetropia, or far-sightedness, occurs when a person can see distant objects clearly but struggles with nearby ones. The near point (normally 25 cm) recedes further away. This happens because the light rays from a nearby object are focused behind the retina. The root cause is either the focal length of the eye lens being too long or the eyeball being too small Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.163. To correct this, we use a converging (convex) lens, which provides the additional refractive power needed to bring the image forward onto the retina.

Presbyopia is often confused with hypermetropia because both affect near vision, but their origins differ. Presbyopia is strictly age-related. As we age, the ciliary muscles weaken and the eye lens loses its flexibility, making it difficult to focus on close objects Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.164. Interestingly, some people develop both myopia (short-sightedness) and presbyopia; they require bifocal lenses, where the upper portion is concave for distance and the lower portion is convex for reading.

Astigmatism arises when the cornea or the eye lens is not perfectly spherical, having different curvatures in different planes. This prevents light from focusing on a single point on the retina, causing blurred vision in all directions. Unlike the spherical lenses used for simple myopia or hypermetropia, astigmatism requires a cylindrical lens to equalize the refraction across different axes.

Defect	Primary Cause	Correction
Hypermetropia	Short eyeball or long focal length	Convex (Converging) lens
Presbyopia	Loss of lens elasticity / weak ciliary muscles	Convex lens or Bifocals
Astigmatism	Irregular curvature of cornea/lens	Cylindrical lens

Sources: Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.162; 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. Optical Instruments in Science and Technology (exam-level)

Optical instruments are sophisticated tools designed to enhance our vision by manipulating light through reflection and refraction. At the most fundamental level, the human eye itself is a biological optical instrument, but it often requires external assistance. When we talk about corrective lenses, we are essentially modifying the eye's refractive power to ensure that light rays converge precisely on the retina. For instance, a short-sighted (myopic) eye cannot see distant objects clearly because the image is formed in front of the retina. To fix this, we use a diverging (concave) lens which spreads the incoming parallel rays so they appear to originate from the person's "far point"—the maximum distance at which they can still see clearly.

The mathematical heart of these instruments lies in the Lens Formula and the concept of Power. The power of a lens (P) is defined as the reciprocal of its focal length (f) in meters: P = 1/f. In the case of myopia, if a person's far point is 200 cm (or 2 meters), we need a lens that creates a virtual image of a distant object at that specific 2-meter mark. Using the lens formula, where the object is at infinity (u = ∞) and the image distance (v) is the far point, the focal length becomes equal to the negative of the far point distance. Thus, for a far point of 2m, f = -2m, and the power is P = 1/(-2) = -0.5 Dioptres (D). The negative sign is a crucial indicator that a concave lens is being used Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.152.

Beyond basic correction, optical instruments like microscopes and telescopes allow us to explore different scales of reality. Simple magnifying glasses or compound microscopes are essential for examining the minute structures of minerals and elements Science, Class VIII NCERT (Revised ed 2025), Nature of Matter: Elements, Compounds, and Mixtures, p.129. Meanwhile, telescopes help us peer into the cosmos. While early designs relied solely on lenses (refracting telescopes), most modern high-performance telescopes are reflecting telescopes. These use large concave mirrors as their primary light-gatherers because mirrors are easier to support and do not suffer from the chromatic aberration (color fringing) common in large lenses Science, Class VIII NCERT (Revised ed 2025), Light: Mirrors and Lenses, p.156.

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.152; Science, Class VIII NCERT (Revised ed 2025), Nature of Matter: Elements, Compounds, and Mixtures, p.129; Science, Class VIII NCERT (Revised ed 2025), Light: Mirrors and Lenses, p.156

6. Myopia and the Mathematics of Lens Power (exam-level)

In our journey through optics, we encounter Myopia, commonly known as near-sightedness. From a biological perspective, a person with myopia can see nearby objects clearly, but distant objects appear blurred because the image is focused in front of the retina rather than directly on it. This typically happens because the eyeball is too long or the eye lens has excessive curvature Science, Class X, The Human Eye and the Colourful World, p.163.

To correct this, we use the mathematics of Lens Power (P). Power is defined as the reciprocal of the focal length (f) expressed in meters (P = 1/f). The SI unit of power is the dioptre (D) Science, Class X, Light – Reflection and Refraction, p.158. For a myopic eye, we need a concave (diverging) lens to spread the light rays before they enter the eye, effectively pushing the focal point back onto the retina.

Feature	Myopic Eye Detail
Far Point	Less than infinity (the maximum distance one can see clearly).
Correction Goal	Form a virtual image of a distant object (at infinity) at the eye's far point.
Lens Formula	1/f = 1/v - 1/u (where u = -∞ and v = -Far Point).
Power Sign	Always negative for myopia correction.

The calculation is elegant in its simplicity: if a person’s far point is 2 meters, the corrective lens must have a focal length (f) of -2 meters. Applying the power formula, P = 1/(-2) = -0.5 D. This negative sign is the standard convention for concave lenses, while positive power indicates a convex lens used for other defects like hypermetropia Science, Class X, Light – Reflection and Refraction, p.159.

Sources: Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.163; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.159

7. Solving the Original PYQ (exam-level)

Now that you have mastered the optics of the human eye, this question brings together three fundamental pillars: sign convention, the lens formula, and the definition of power. In myopia (short-sightedness), the eye's far point has moved from infinity to a closer distance—in this case, 200 cm. To correct this, we must use a diverging (concave) lens that takes an object at infinity and forms a virtual image at the person’s new far point. As per the NCERT Science Class 10, since the image is formed on the same side as the object, the image distance (v) is -200 cm, which we must convert to -2 meters for power calculations.

To find the power, we first determine the focal length (f). Using the lens formula where the object distance (u) is at infinity, the focal length is simply equal to the image distance (v), which is the far point: f = -2 m. The Power (P) of a lens is defined as the reciprocal of its focal length expressed specifically in meters (P = 1/f). Substituting our value, we get P = 1/(-2), which equals -0.5 Dioptres (D). The negative sign is the diagnostic hallmark of a concave lens, confirming we have corrected the myopia correctly. Thus, (A) -0.5D is our correct answer.

UPSC often sets traps using unit conversions and sign conventions. Option (B) 2D is a common mistake where a student might forget the reciprocal or use the focal length in meters directly as the power. Options (C) and (D) are distractors for those who might miscalculate the fraction or fail to recognize that a negative power is mandatory for short-sightedness. Always remember: positive power is for convex lenses (Hypermetropia), and negative power is for concave lenses (Myopia). Precision in these basic conventions is what ensures success in Science & Technology questions.