The ratio of the focal length of the objective to the focal length of the eyepiece is greater than one for

1. Basics of Refraction and Lens Types (basic)

Welcome to your journey into Geometrical Optics! To understand how complex instruments like telescopes work, we must first master the Basics of Refraction. Refraction is the phenomenon where light changes its direction when it travels obliquely from one transparent medium to another. This bending occurs because the speed of light varies depending on the material it is passing through Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.134.

When we apply this principle to Spherical Lenses, we can control how light behaves. A lens is a transparent material bound by two surfaces, at least one of which is spherical. Depending on their shape, lenses interact with light in two distinct ways:

• Convex Lens: Thicker in the middle than at the edges. It is called a converging lens because parallel rays of light passing through it meet at a single point called the focus Science, Class VIII, NCERT (Revised ed 2025), Light: Mirrors and Lenses, p.164.
• Concave Lens: Thinner in the middle than at the edges. It is called a diverging lens because it spreads parallel light rays apart, making them appear as if they are coming from a point behind the lens.

The distance from the center of the lens to this focus point is known as the focal length (f). This brings us to a critical concept: Power (P). The power of a lens represents its ability to converge or diverge light. Mathematically, it is the reciprocal of the focal length (measured in meters), expressed as P = 1/f Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.157. A shorter focal length means the lens bends light more sharply, thus having a higher power.

Feature	Convex Lens	Concave Lens
Physical Shape	Bulging outwards	Curved inwards
Effect on Light	Converges (Brings together)	Diverges (Spreads out)
Nature of Image	Can form real or virtual images	Always forms virtual, diminished images

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.134, 157; Science, Class VIII, NCERT (Revised ed 2025), Light: Mirrors and Lenses, p.164

2. Understanding Focal Length and Optical Power (basic)

To understand how complex optical instruments like telescopes or microscopes work, we must first master the two most fundamental properties of any lens or mirror: Focal Length and Optical Power. Think of these as the 'strength' and 'character' of the lens.

The Focal Length (f) is the distance between the center of the lens (the optical center) and the point where parallel rays of light actually meet or appear to meet after passing through the lens. This meeting point is called the Principal Focus Science, Class X, p.151. In spherical mirrors, this distance is measured from the pole of the mirror Science, Class X, p.136. Essentially, the focal length tells us how 'sharply' a lens bends light. A lens with a very short focal length is highly curved and bends light drastically, while a long focal length lens is flatter and bends light more gradually.

This leads us to the concept of Optical Power (P). Power is simply a measure of a lens's ability to converge or diverge light rays. Mathematically, it is the reciprocal of the focal length, expressed as P = 1/f. The standard unit for power is the dioptre (D), and it is crucial to remember that this formula only works when the focal length is measured in metres Science, Class X, p.158. A lens that can focus light within just 0.5 metres is more 'powerful' (2D) than one that takes 1.0 metre to focus light (1D).

Lens Type	Nature of Light	Focal Length (f) / Power (P)
Convex Lens	Converging	Positive (+)
Concave Lens	Diverging	Negative (–)

In practice, opticians use these signs to communicate. If a doctor prescribes a lens with +2.0 D, they are telling the lab to provide a convex lens with a focal length of +0.50 m Science, Class X, p.158. Understanding this relationship is the secret to understanding how lenses are combined to build powerful scientific tools.

Sources: Science, Class X, Light – Reflection and Refraction, p.136; Science, Class X, Light – Reflection and Refraction, p.151; Science, Class X, Light – Reflection and Refraction, p.158

3. The Human Eye and Vision Correction (intermediate)

To understand vision, we must first look at the Power of Accommodation. Unlike a camera where the lens moves physically to focus, the human eye uses ciliary muscles to change the curvature—and thus the focal length—of the crystalline lens. This remarkable biological adjustment allows us to see both distant stars and a nearby book clearly Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.170. For a healthy young adult, the near point (the closest distance for clear vision without strain) is about 25 cm, while the far point is theoretically at infinity Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.164.

When the eye cannot properly focus images on the retina, we encounter refractive defects. These are primarily caused by issues with eyeball length or lens flexibility. We categorize them as follows:

Defect	Problem	Image Formation	Correction
Myopia (Near-sightedness)	Excessive curvature of the lens or long eyeball.	In front of the retina.	Concave lens (Diverging) Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.170
Hypermetropia (Far-sightedness)	Focal length is too long or eyeball is too short.	Behind the retina.	Convex lens (Converging) Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.162
Presbyopia	Aging ciliary muscles; loss of lens flexibility.	Difficulty focusing on nearby objects.	Bifocal lenses (often both convex and concave) Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.163

In clinical practice, these corrections are prescribed using Lens Power (P), measured in Dioptres (D), where P = 1/f (focal length in meters). A negative power (e.g., -5.5 D) indicates a concave lens for Myopia, while a positive power (e.g., +1.5 D) indicates a convex lens for Hypermetropia Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.170. Understanding this relationship is vital for mastering how light is manipulated by both biological and artificial optical systems.

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; Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.170

4. Atmospheric Optics and Total Internal Reflection (intermediate)

To understand Atmospheric Optics, we must first master the boundary behavior of light. When light travels from an optically denser medium (like water or glass) to a rarer medium (like air), it bends away from the normal. As the angle of incidence increases, the refracted ray gets closer and closer to the interface. Eventually, we reach the critical angle—the specific angle of incidence where the refracted ray travels exactly along the boundary. If the incident angle increases even slightly beyond this, the light can no longer escape the denser medium; instead, it reflects back entirely. This phenomenon is known as Total Internal Reflection (TIR), a concept rooted in the fundamental laws of refraction Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.134.

In our atmosphere, the air is rarely uniform. Temperature differences create layers of varying density, meaning the refractive index changes with altitude. This leads to Atmospheric Refraction. On a hot summer day, the air near the asphalt is much hotter (and thus rarer) than the air above it. Light from the distant sky or objects travels downward from cooler, denser air into the hotter, rarer layers. As it does, it bends increasingly away from the normal until it undergoes TIR near the ground. To your eyes, this reflected light appears to come from the ground itself, creating the illusion of a pool of water reflecting the sky—this is the classic Mirage. This is why we say the atmosphere acts like a lens with a varying refractive index Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.165.

Beyond mirages, TIR is the physical backbone of modern communication. Optical fibers use a high-refractive-index core surrounded by a lower-index cladding to trap light signals through a series of continuous total internal reflections. This allows data to travel vast distances with minimal loss. Whether it's the sparkling of a diamond (caused by its high refractive index and low critical angle) or the twinkling of stars (caused by fluctuating atmospheric refraction), these phenomena remind us that light rarely travels in a perfectly simple path when density is in flux.

Feature	Refraction	Total Internal Reflection (TIR)
Medium Transition	Any change in optical density.	Must move from Denser to Rarer medium.
Angle Requirement	Any angle of incidence.	Incidence angle must be greater than the critical angle.
Light Loss	Some light is reflected, some is refracted.	100% of light is reflected back; no energy is lost to refraction.

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.134; Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.165

5. Principles of Angular Magnification (intermediate)

When we look at distant stars or tiny bacteria, linear magnification (the ratio of image height to object height) isn't always the most helpful metric. After all, a star's physical height is millions of kilometers, so a ratio doesn't tell us how "large" it looks to our eye. Instead, we use angular magnification. This principle measures how much larger the angle subtended by the image at our eye is compared to the angle subtended by the object itself. While standard lenses are often described by the ratio m = v/u Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.156, optical instruments like telescopes and microscopes rely on the relationship between the focal lengths of their two primary components: the objective and the eyepiece.

In a telescope, the goal is to view distant objects. The objective lens (or mirror) has a long focal length (fₒ) to capture light from infinity and form an image at its focus. This image is then viewed through an eyepiece with a short focal length (fₑ). The angular magnification (M) is determined by the ratio M = fₒ / fₑ. To achieve high magnification, designers maximize this ratio by making the objective focal length very long (e.g., 1000mm) and the eyepiece focal length very short (e.g., 10mm). This is why professional telescopes are often very long tubes!

In contrast, a compound microscope operates on a different logic. Because the object is very close, the objective lens must have a very short focal length to produce a highly magnified real image. The eyepiece then acts as a simple magnifier to further enlarge this image. Consequently, the design requirements for the ratio of focal lengths differ significantly between the two instruments:

Feature	Telescope	Compound Microscope
Objective Focal Length (fₒ)	Large/Long	Very Small/Short
Eyepiece Focal Length (fₑ)	Small/Short	Relatively larger than fₒ
Ratio fₒ / fₑ	Typically > 1	Typically < 1

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.156

6. The Compound Microscope: Design and Focal Lengths (exam-level)

In our journey through optics, we have seen how a single lens can magnify an object. However, to see the intricate details of tiny specimens—like the microscopic components of a mixture that remain invisible to the naked eye—we require a compound microscope Science, Class VIII NCERT, Nature of Matter, p.117. Unlike a simple magnifying glass, the compound microscope uses a system of lenses to achieve much higher magnification and minimize optical defects that occur with a single lens Science, Class X NCERT, Light – Reflection and Refraction, p.158.

The design consists of two main converging lenses: the objective (facing the object) and the eyepiece (facing the eye). Each lens has two principal foci, and the distance from the optical center to these points is the focal length (f) Science, Class X NCERT, Light – Reflection and Refraction, p.151. In a microscope, the objective lens has an extremely short focal length (fₒ) and a small aperture. This allows it to be placed very close to the specimen, creating a highly magnified real image inside the microscope tube. The eyepiece, which has a relatively larger focal length (fₑ), then acts as a second magnifier to further enlarge this image for the observer.

A common point of confusion in competitive exams is the comparison between microscopes and telescopes. The fundamental difference lies in the ratio of their focal lengths. In a telescope, the objective must be large and have a long focal length to capture light from distant stars, meaning fₒ > fₑ. Conversely, in a compound microscope, the objective is purposefully kept much shorter than the eyepiece to maximize the initial stage of magnification. Therefore, for a microscope, the ratio fₒ / fₑ < 1.

Feature	Compound Microscope	Astronomical Telescope
Objective Focal Length (fₒ)	Very Short	Very Long
Eyepiece Focal Length (fₑ)	Relatively Larger than fₒ	Short
Focal Length Ratio	fₒ < fₑ	fₒ > fₑ

Sources: Science, Class VIII NCERT, Nature of Matter, p.117; Science, Class X NCERT, Light – Reflection and Refraction, p.151; Science, Class X NCERT, Light – Reflection and Refraction, p.158

7. The Astronomical Telescope: Design and Ratios (exam-level)

To understand the design of an astronomical telescope, we must first look at its core purpose: to make distant, massive objects like planets or stars appear larger to our eyes. This is achieved by increasing the angular magnification (or magnifying power). A telescope consists of two primary optical elements: the objective (the lens or mirror facing the object) and the eyepiece (the lens the observer looks through). While modern large-scale telescopes often use mirrors to avoid weight and clarity issues Science, Grade VIII, p.156, the fundamental principle of focal length ratios remains the same.

The magnifying power (M) of a telescope in normal adjustment is defined by the ratio of the focal length of the objective (fₒ) to the focal length of the eyepiece (fₑ): M = fₒ / fₑ. To achieve high magnification, designers use an objective with a very long focal length and an eyepiece with a very short focal length. This ensures that the ratio fₒ / fₑ is significantly greater than 1. This is the opposite of a compound microscope, where the objective has a very short focal length to magnify a nearby specimen. In a telescope, a large objective lens also helps in "light gathering," allowing us to see faint stars that are otherwise invisible.

Feature	Astronomical Telescope	Compound Microscope
Objective Focal Length (fₒ)	Long (to capture distant light)	Very Short (to be near the object)
Eyepiece Focal Length (fₑ)	Short	Short (but usually fₑ > fₒ)
Ratio fₒ / fₑ	Greater than 1 (fₒ > fₑ)	Less than 1 (fₒ < fₑ)

It is also important to note that lens systems in these instruments often use a combination of lenses rather than a single piece of glass. This additive property of lens power helps minimize optical defects and improves image sharpness Science, Class X, p.158. When you look through a telescope, the objective forms a real image of the distant star, and the eyepiece then acts as a simple magnifier to enlarge that image for your eye Science, Class X, p.138.

Sources: Science, Class VIII, Beyond Earth, p.156; Science, Class X, Light – Reflection and Refraction, p.138; Science, Class X, Light – Reflection and Refraction, p.158

8. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental properties of lenses and the sign convention, this question asks you to apply those building blocks to the functional design of optical instruments. The core concept here is angular magnification. In a telescope, the primary goal is to resolve distant, massive objects by increasing the angle they subtend at the eye. As you learned in the derivation for magnifying power, $M = f_o / f_e$. To achieve a high magnification for celestial observation, the objective lens must have a very long focal length to capture light from infinity, while the eyepiece acts as a short-focal-length magnifier. Therefore, for (B) a telescope, the ratio $f_o / f_e$ must be greater than one.

UPSC frequently tests whether you can distinguish between instruments that seem similar but serve opposite optical purposes. In a compound microscope, the objective lens is designed to produce a large linear magnification of a tiny, nearby specimen. To do this, the objective must have an extremely short focal length ($f_o$), often much smaller than that of the eyepiece ($f_e$). Consequently, the ratio $f_o / f_e$ for a microscope is typically less than one, making option (A) incorrect. Option (C) is a common trap meant to catch students who recall that both instruments use two lenses but forget that their optical geometries are fundamentally inverted, as detailed in NCERT Class 12 Physics: Optics.