Two thin convex lenses of focal lengths 4 cm and 8 cm are separated by a distance of 4 cm in air. The combination will have the focal length

1. Basics of Refraction and Refractive Index (basic)

Imagine light as a fast-moving traveler that suddenly encounters a change in terrain—moving from a smooth highway to a sandy beach. Naturally, its speed changes, and if it hits the sand at an angle, its direction shifts. This bending of light as it passes obliquely from one transparent medium to another is called refraction. At its heart, refraction is a consequence of light changing its speed as it enters a new material Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.147.

To understand this precisely, we look at the Laws of Refraction. 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, 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 (NCERT 2025 ed.), Light – Reflection and Refraction, p.148. This constant is what we call the Refractive Index (n).

The Refractive Index is a measure of how much a medium slows down light. It can be expressed in two ways:

• Relative Refractive Index (n₂₁): The ratio of the speed of light in medium 1 (v₁) to the speed of light in medium 2 (v₂). Formula: n₂₁ = v₁ / v₂.
• Absolute Refractive Index (n): When the first medium is vacuum (or air), we compare the speed of light in vacuum (c ≈ 3 × 10⁸ m/s) to the speed in the medium (v). Formula: n = c / v.

A very important distinction to keep in mind is the difference between optical density and mass density. A medium with a higher refractive index is "optically denser," meaning light travels slower through it. Interestingly, an optically denser medium might not have a higher mass density; for instance, kerosene has a higher refractive index (1.44) than water (1.33), even though kerosene is less dense than water and floats on it Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, 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

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.147-149; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158

2. Understanding Convex and Concave Lenses (basic)

A lens is a piece of transparent material, such as glass, bound by two surfaces, where at least one surface is spherical. Depending on how these surfaces are curved, we classify them into two primary types: convex and concave. A convex lens (or double convex lens) is thicker at the middle than at the edges. Conversely, a concave lens (or double concave lens) is thinner in the middle and thicker at the edges Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.150.

The defining characteristic of these lenses is how they manipulate parallel rays of light. When parallel light rays pass through a convex lens, they bend inward and meet at a single point called the principal focus (F). Because of this property, a convex lens is also known as a converging lens. On the other hand, a concave lens causes parallel rays to bend outward, making them appear to diverge from a point on the same side as the incoming light. For this reason, it is known as a diverging lens. Every lens has two principal foci, one on each side, denoted as F₁ and F₂ Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.151.

To understand the strength or reach of a lens, we measure the focal length (f), which is the distance between the optical center (the central point of the lens) and the principal focus. In practical terms, a convex lens can form both real and virtual images depending on the object's distance, often used to magnify small text. A concave lens, however, always produces a virtual, erect, and diminished (smaller) image, regardless of where the object is placed Science, Class VIII (NCERT Revised ed. 2025), Light: Mirrors and Lenses, p.163.

Feature	Convex Lens	Concave Lens
Shape	Thicker in the middle	Thicker at the edges
Light Action	Converging (rays meet)	Diverging (rays spread)
Common Use	Magnifying glass, Reading small print	Correcting nearsightedness

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 VIII (NCERT Revised ed. 2025), Light: Mirrors and Lenses, p.163

3. Power of a Lens and the Dioptre (basic)

When we talk about the Power of a Lens, we are describing its ability to bend light rays. Simply put, a lens with a shorter focal length bends light more sharply, meaning it has a greater "power" to converge or diverge light. For example, a thick convex lens with a short focal length will converge light rays much more strongly than a thinner one. This inverse relationship is the foundation of the concept: power is defined as the reciprocal of the focal length.

The SI unit of power is the dioptre (denoted by D). It is critical to remember that for the power to be expressed in dioptres, the focal length (f) must be measured in metres (m). Therefore, the formula is P = 1/f. One dioptre is defined as the power of a lens whose focal length is exactly 1 metre (Science, Class X, Light – Reflection and Refraction, p.158). If you are given the focal length in centimetres, you must convert it to metres before calculating the power (e.g., 25 cm = 0.25 m).

In practice, opticians use sign conventions to distinguish between lens types. Since a convex lens has a positive focal length, its power is always positive (+). Conversely, a concave lens has a negative focal length, so its power is negative (-) (Science, Class X, Light – Reflection and Refraction, p.158). This is why a prescription for "-2.5 D" immediately tells a doctor that the patient requires a diverging (concave) lens for myopia (Science, Class X, The Human Eye and the Colourful World, p.170).

When multiple thin lenses are placed in contact with each other, their net power is simply the algebraic sum of their individual powers (P = P₁ + P₂ + ...). However, if the lenses are separated by a distance d, we must account for that gap. The effective focal length (F) for two lenses separated by distance 'd' is calculated as:
1/F = 1/f₁ + 1/f₂ - d/(f₁f₂). This formula shows that increasing the distance between lenses reduces the overall converging power of the system.

Sources: Science, Class X, Light – Reflection and Refraction, p.158; Science, Class X, The Human Eye and the Colourful World, p.170

4. Human Eye: Vision Defects and Lens Correction (intermediate)

The human eye is a biological marvel that functions much like a camera, but with a sophisticated power of accommodation. This is the ability of the ciliary muscles to modify the curvature of the eye lens, thereby adjusting its focal length to focus clearly on both nearby and distant objects Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.170. For a healthy young adult, the least distance of distinct vision (near point) is approximately 25 cm. However, when the eye loses this flexibility or when the eyeball's shape is slightly off, we encounter refractive defects. There are two primary structural defects: Myopia and Hypermetropia. In Myopia (near-sightedness), the person can see nearby objects but distant ones appear blurred because the image is formed in front of the retina. Conversely, in Hypermetropia (far-sightedness), distant objects are clear, but nearby light rays are focused behind the retina Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.163. Correction involves placing a second lens in front of the eye to shift the final image position. The physics here is based on the effective focal length: when you add a corrective lens of focal length f₂ to your eye lens f₁, the combined system's power changes to bring the focal point precisely onto the retina.

Feature	Myopia (Near-sightedness)	Hypermetropia (Far-sightedness)
Image Formation	In front of the retina	Behind the retina
Possible Causes	Eyeball too long or excessive lens curvature	Eyeball too short or lens focal length too long
Correction	Concave lens (Diverging)	Convex lens (Converging)

As we age, we may also develop Presbyopia. This occurs because the ciliary muscles weaken and the eye lens loses its elasticity, causing the near point to recede Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.164. Some individuals suffer from both defects simultaneously and require bi-focal lenses, where the upper part (concave) assists with distance vision and the lower part (convex) assists with reading.

Sources: Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.162-164, 170

5. Optical Instruments: Telescopes and Microscopes (intermediate)

To understand complex optical instruments like microscopes and telescopes, we must first look at how lenses work together. While a single convex lens can act as a simple magnifier, it has physical limitations—it can only magnify so much before the image becomes distorted. To overcome this, we use a combination of lenses. When lenses are placed in direct contact, their net power (P) is simply the algebraic sum of their individual powers: P = P₁ + P₂. This additive property allows engineers to design lens systems that minimize optical defects Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158.

However, in instruments like the compound microscope or the astronomical telescope, the lenses are not touching; they are separated by a specific distance (d) within a tube. This separation changes the math. The effective focal length (F) of two lenses with focal lengths f₁ and f₂ separated by a distance d is calculated using the formula:

1/F = 1/f₁ + 1/f₂ - d/(f₁f₂)

In a microscope, we use an objective lens (near the object) and an eyepiece (near the eye). The objective forms a real, inverted image, which then acts as the 'object' for the eyepiece, resulting in a much larger final image than a single lens could ever produce Science, Class VIII (NCERT 2025 ed.), Nature of Matter: Elements, Compounds, and Mixtures, p.117.

Instrument	Primary Purpose	Lens Configuration
Microscope	Magnify tiny, near objects	Short focal length objective and eyepiece
Telescope	Magnify distant, large objects	Large aperture objective to collect more light

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158; Science, Class VIII (NCERT 2025 ed.), Nature of Matter: Elements, Compounds, and Mixtures, p.117

6. Equivalent Focal Length of Lenses in Contact (intermediate)

In optical engineering, we rarely use a single lens. To create high-quality images in cameras or to correct complex vision defects, we combine multiple lenses. The most fundamental concept here is Lens Power (P), which is the reciprocal of focal length (P = 1/f). When thin lenses are placed in direct contact, their total power is simply the algebraic sum of their individual powers: P = P₁ + P₂ + .... This additive property allows doctors to precisely tailor corrective eyewear by stacking lenses of different strengths Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158.

For two lenses with focal lengths f₁ and f₂ in contact, the equivalent focal length (F) is calculated as: 1/F = 1/f₁ + 1/f₂. However, if the lenses are separated by a distance (d), the light rays from the first lens travel through a gap before hitting the second, which alters the final convergence. In this case, we use the generalized formula:
1/F = 1/f₁ + 1/f₂ - d/(f₁f₂)
This formula shows that the separation distance reduces the total power of the system (since power is subtracted). It is important to always use the correct sign convention: focal length is positive for a convex (converging) lens and negative for a concave (diverging) lens Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.160.

This principle is applied in bi-focal lenses, which combine a concave portion for distant vision and a convex portion for near vision, allowing a person suffering from both myopia and hypermetropia to see clearly at all distances Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.164. By adjusting the distance between lenses in a telescope or camera zoom lens, engineers can vary the effective focal length of the entire system without changing the physical lenses themselves.

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.160; Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.164

7. Lens Combinations Separated by a Distance (exam-level)

In our journey through optics, we have seen how a single lens converges or diverges light based on its focal length Science, Light – Reflection and Refraction, p.151. However, sophisticated optical instruments like telescopes, microscopes, and high-end camera lenses rarely rely on a single lens. Instead, they use a combination of lenses. When these lenses are not touching but are separated by a finite distance (d), the way they bend light changes significantly compared to when they are in contact.

The effective focal length (F) of two thin lenses with focal lengths f₁ and f₂, separated by a distance d, is given by the following relationship:

1/F = 1/f₁ + 1/f₂ - d/(f₁f₂)

Notice the third term in this equation: - d/(f₁f₂). This is the "separation factor." In a system of two convex (converging) lenses, this term is negative, which means that increasing the distance between the lenses actually reduces the total converging power of the system. This principle is fundamental in zoom lenses, where moving lens elements closer or further apart changes the effective focal length of the entire camera assembly.

When applying this formula, it is critical to adhere to the sign convention Science, Light – Reflection and Refraction, p.155. For instance, if one lens is concave (diverging), its focal length must be entered as a negative value. The formula can also be expressed in terms of Optical Power (P), where P = 1/f. The equivalent power is P = P₁ + P₂ - dP₁P₂. This illustrates that the total power of the combination is not just a simple sum when a gap exists between the lenses.

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

8. Solving the Original PYQ (exam-level)

Now that you have mastered the behavior of individual thin lenses, this question challenges you to apply the principle of a lens system. While you previously learned how powers add up when lenses are in direct contact, here we introduce separation distance (d). The fundamental concept to bridge here is that the effective power of a combination changes when lenses are spaced apart, as the divergence or convergence from the first lens hits the second lens at a different height. This necessitates the use of the equivalent focal length formula for separated lenses.

To arrive at the result, we walk through the formula: 1/F = 1/f₁ + 1/f₂ - d/(f₁f₂). By substituting the given values—f₁ = 4 cm, f₂ = 8 cm, and d = 4 cm—the calculation becomes 1/4 + 1/8 - 4/(4 × 8). It is important to observe how the term 4/32 simplifies to 1/8. In this specific scenario, the subtractive effect of the distance exactly cancels out the contribution of the second lens (+1/8 - 1/8), leaving us with 1/F = 1/4. Therefore, the correct answer is (A) 4 cm.

UPSC often includes options to catch students who use "shortcuts" that ignore physical realities. For instance, Option (C) 12 cm is a trap for those who simply add the focal lengths (4 + 8), which is never correct in optics. Option (B) 8 cm assumes the first lens has no effect at all. As discussed in NCERT Class 12 Physics (Ray Optics), the separation distance effectively reduces the total converging power of the system compared to if they were in contact; missing that subtractive term is the most common reason for error in these competitive problems.