The focal length of a convex lens is

1. Basics of Refraction and Snell's Law (basic)

Welcome to your first step in mastering Geometrical Optics! To understand how lenses work, we must first master Refraction. In simple terms, refraction is the change in direction of light as it passes obliquely from one transparent medium into another. This happens because light travels at different speeds in different materials. While light travels at its maximum speed in a vacuum (approximately 3 × 10⁸ m/s), it slows down when it enters media like water or glass Science, Class X, Light – Reflection and Refraction, p.159.

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. The second, and perhaps most famous, is Snell’s Law. It states that for a given pair of media and a specific color of light, the ratio of the sine of the angle of incidence (i) to the sine of the angle of refraction (r) is a constant Science, Class X, Light – Reflection and Refraction, p.148. This relationship is expressed as:

sin i / sin r = constant (n₂₁)

This constant is known as the Refractive Index of the second medium relative to the first. It is a crucial measure of a medium's "optical density." The higher the refractive index, the more the light slows down and the more it bends toward the normal. Note that optical density is not the same as mass density; for example, kerosene has a higher refractive index than water, even though it is less dense and floats on it.

Scenario	Speed Change	Bending Direction
Rarer to Denser (e.g., Air to Glass)	Light slows down	Bends towards the normal
Denser to Rarer (e.g., Glass to Air)	Light speeds up	Bends away from the normal

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.147; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.159

2. Terminology of Lenses: Focus and Focal Length (basic)

To understand how lenses work, we must first look at the Principal Focus (F). Imagine a bundle of light rays parallel to the Principal Axis (the imaginary line passing through the center of the lens). When these rays hit a Convex lens, they are bent inwards and meet at a single point on the other side. This point is the Principal Focus. Conversely, a Concave lens bends rays outwards; if you trace these diverging rays backward, they appear to originate from a point, which we also call the focus Science, Class X, p.150. Since light can pass through a lens from either side, every lens has two principal foci, usually denoted as F₁ and F₂.

The distance from the Optical Centre (the geometric center of the lens) to the Principal Focus is known as the Focal Length (f) Science, Class X, p.151. This length is a measure of the lens's power to converge or diverge light. For example, a thicker convex lens bends light more sharply, resulting in a shorter focal length compared to a thinner lens. This is why you would choose a lens with a small focal length, like 5 cm, to see tiny dictionary letters clearly — it provides higher magnification Science, Class X, p.160.

Interestingly, the focal length is not just a property of the lens's shape; it also depends on the light itself. According to the Lens Maker's Formula, the focal length is inversely proportional to the refractive index (n) of the lens material. Because different colors of light have different wavelengths, the lens treats them differently. Blue light has a shorter wavelength and experiences a higher refractive index, causing it to bend more sharply toward the axis. Red light, with a longer wavelength, bends less. Therefore, blue light converges closer to the lens than red light, meaning the focal length for blue light is slightly shorter than for red light.

Feature	Convex Lens	Concave Lens
Nature	Converging lens	Diverging lens
Focus	Real (rays actually meet)	Virtual (rays appear to meet)
Focal Length	Varies by color (Shortest for Blue)	Varies by color (Shortest for Blue)

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.160

3. The Visible Spectrum and Dispersion (basic)

To understand why a simple lens can sometimes produce a 'blurry' image with colored edges, we must first understand the nature of white light. White light is not a single entity but a mixture of seven constituent colors: Violet, Indigo, Blue, Green, Yellow, Orange, and Red (VIBGYOR). When white light enters a transparent medium like a glass prism, it undergoes dispersion—the process of splitting into its component colors Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.167. This happens because each color of light has a different wavelength, and the medium (glass) reacts differently to each one.

The 'reaction' of the glass is measured by its refractive index (n). Interestingly, the refractive index is not a fixed constant for a material; it actually varies based on the wavelength of light passing through it. This relationship is described by Cauchy's equation, which tells us that the refractive index is higher for light with shorter wavelengths. Consequently, violet and blue light (short wavelength) experience a higher refractive index and bend more sharply than red light (long wavelength), which bends the least Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.167.

When we apply this to a convex lens, the implications are profound. Since a lens is essentially a sophisticated refracting tool, it bends blue light more aggressively toward the optical axis than it does red light. Following the Lens Maker's Formula, where focal length (f) is inversely proportional to (n-1), a higher refractive index results in a shorter focal length. Therefore, blue light converges at a point closer to the lens, while red light converges further away. This discrepancy in focal points for different colors is known as chromatic aberration, and it explains why images through simple lenses often have color fringes.

Property	Violet / Blue Light	Red Light
Wavelength	Shorter	Longer
Refractive Index (n)	Higher	Lower
Bending (Deviation)	More pronounced	Least pronounced
Focal Length (f)	Shorter (closer to lens)	Longer (further from lens)

Sources: Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.167; Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.166

4. Applications of Lenses: Correcting Vision (exam-level)

To understand how we correct vision, we must first look at the eye as a biological camera. The primary goal of the eye is to focus light precisely onto the retina. When the eye’s power of accommodation—its ability to adjust the focal length of the crystalline lens—diminishes, vision becomes blurred Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.162. We correct these 'refractive defects' by placing an external spherical lens in front of the eye to modify the incoming light's path before it enters the eye.

There are three primary defects of vision we address with lenses:

• Myopia (Near-sightedness): The eye converges light too strongly, forming the image in front of the retina. We use a concave (diverging) lens to spread the light out slightly before it hits the eye, pushing the image back onto the retina.
• Hypermetropia (Far-sightedness): The eye converges light too weakly, so the image would theoretically form behind the retina. We use a convex (converging) lens to provide extra 'bending power' and bring the image forward.
• Presbyopia: Often occurring with age, the ciliary muscles weaken, making it hard to see nearby objects clearly. This often necessitates bi-focal lenses, where 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.

The strength of these corrective lenses is measured in Power (P), which is the reciprocal of the focal length (f) in metres (P = 1/f). The SI unit is the dioptre (D). Crucially, we use a sign convention: a convex lens has a positive power, while a concave lens has a negative power Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158. For example, a prescription of –2.0 D indicates a concave lens with a focal length of –0.5 m, designed to treat myopia.

Defect	Focus Point	Corrective Lens	Power Sign
Myopia	In front of retina	Concave (Diverging)	Negative (–)
Hypermetropia	Behind retina	Convex (Converging)	Positive (+)

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.164; Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158

5. Refractive Index and its Wavelength Dependence (intermediate)

To understand how light behaves in a lens, we must first look at the Refractive Index (n). At its simplest, the refractive index is a measure of how much a medium slows down light compared to its speed in a vacuum. It is defined by the ratio n = c/v, where c is the speed of light in vacuum and v is the speed in the medium Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148. While we often treat the refractive index of glass as a single value (like 1.50), it actually varies slightly depending on the wavelength (color) of the light passing through it.

This variation is governed by Cauchy’s Equation, which tells us that the refractive index is inversely related to the wavelength. In practical terms: shorter wavelengths experience a higher refractive index. Since blue light has a shorter wavelength than red light, the glass "feels" denser to blue light than it does to red light. Consequently, blue light slows down more and undergoes a greater degree of refraction (bending) as it enters the lens material Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.159.

This difference in bending has a direct impact on the focal length (f) of a lens. According to the Lens Maker’s Formula, the power of a lens is proportional to (n - 1). Because blue light has a higher refractive index (n), the lens acts with more "power" on blue rays, bending them more sharply toward the optical axis. This causes blue light to converge at a point closer to the lens, resulting in a shorter focal length for blue light compared to red light. This mismatch in focal points for different colors is the fundamental cause of chromatic aberration, where images may appear with colored blurry edges.

Color	Wavelength (λ)	Refractive Index (n)	Bending Angle	Focal Length (f)
Red	Longer	Lower	Less	Longer (Farther)
Blue	Shorter	Higher	More	Shorter (Closer)

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.159

6. The Lens Maker's Formula (intermediate)

While the Lens Formula describes the relationship between object distance (u), image distance (v), and focal length (f) (Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.155), the Lens Maker's Formula gets to the heart of how a lens is actually constructed. It tells us that the focal length is determined by two main factors: the refractive index (n) of the lens material and the radius of curvature (R) of its two surfaces. Mathematically, it is expressed as: 1/f = (n - 1)(1/R₁ - 1/R₂). From this, we can see that the focal length is inversely proportional to (n - 1). The more a material can bend light (higher n), the shorter the focal length becomes.
However, there is a fascinating twist: the refractive index is not a fixed number for a specific material. According to Cauchy's equation, the refractive index varies with the wavelength (λ) of light. Shorter wavelengths, like blue light, experience a higher refractive index than longer wavelengths, like red light. Because blue light is 'slowed down' and bent more sharply by the glass, it converges at a point closer to the lens. This means the focal length for blue light is shorter than the focal length for red light. In a perfect lens, we want all colors to meet at one point, but this variation leads to a common distortion known as chromatic aberration.
Understanding this dependency is vital because it reminds us that light is not just a single ray, but a spectrum. When we calculate the power of a lens in dioptres (D) (Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158), we are usually using an average refractive index, but precision optical instruments must account for these subtle color-based differences to ensure a sharp, clear image.

Color	Wavelength (λ)	Refractive Index (n)	Bending (Refraction)	Focal Length (f)
Red	Longer	Lower	Less	Longer
Blue	Shorter	Higher	More	Shorter

Sources: Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.151, 155, 158

7. Chromatic Aberration in Lenses (exam-level)

To understand Chromatic Aberration, we must first revisit how a lens works. As we know, a lens forms images by refracting (bending) light rays Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.152. However, a single lens made of one material cannot focus all colors of white light at the same point. This phenomenon occurs because the refractive index (n) of a material is not a fixed number—it varies depending on the wavelength (λ) of the light passing through it. According to the Lens Maker's Formula, the focal length (f) is inversely proportional to (n - 1). This means that if the refractive index increases, the focal length decreases (the lens becomes 'stronger' and bends light more). Based on Cauchy’s Equation, the refractive index is higher for shorter wavelengths. Since blue/violet light has a shorter wavelength than red light, it 'sees' the lens as being optically denser. Consequently, blue light is refracted more sharply and converges at a point closer to the lens than red light.

Light Color	Wavelength (λ)	Refractive Index (n)	Bending Power	Focal Length (f)
Blue/Violet	Shorter	Higher	Stronger	Shorter (closer to lens)
Red	Longer	Lower	Weaker	Longer (farther from lens)

This separation of colors means that instead of a single, sharp focus, the lens produces a series of colored images along the optical axis. This results in an image with blurred edges and colored fringes (halos), which can significantly degrade the quality of optical instruments like telescopes or cameras. While a double convex lens is excellent for converging light Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.150, this 'dispersion' of colors is an inherent defect that engineers must correct using combinations of different glass types.

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.150; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.152

8. Solving the Original PYQ (exam-level)

Now that you have mastered the building blocks of optics, this question serves as the perfect synthesis of the Lens Maker's Formula and Cauchy's Equation. To solve this, you must connect the dots between wavelength, refractive index, and the physical bending of light. In your previous lessons, you learned that a lens doesn't just have one fixed focal length for all scenarios; rather, the focal length is inversely proportional to the refractive index (n) of the material. This means that as the material's ability to bend light increases, the point of convergence (focal length) moves closer to the lens.

To arrive at the correct answer, follow this logical chain: First, remember the VIBGYOR spectrum where blue light has a shorter wavelength than red light. According to Cauchy's Relation, the refractive index is higher for shorter wavelengths. Because the lens "sees" blue light as having a higher refractive index, it refracts (bends) blue light more aggressively toward the optical axis. Consequently, blue light converges sooner, making the focal length shorter for blue light than for red. This phenomenon is precisely what causes chromatic aberration, where different colors fail to focus at the same point.

UPSC often uses "reversal traps" to test your precision. Option (C) is the most common pitfall; it acknowledges a difference but flips the relationship, betting that a student might confuse bending power with focal distance. Option (A) is a conceptual trap for those who overlook dispersion, assuming a lens is a static object. Finally, Option (D) is a distractor involving yellow light, which is often used as the "mean" wavelength in optics calculations but does not represent a physical maximum or minimum in this context.