The radius of curvature of a plane mirror

1. Foundations of Light and Laws of Reflection (basic)

At its most fundamental level, light behaves as if it travels in straight lines, a concept that forms the bedrock of geometrical optics Science, Class X (NCERT 2025 ed.), Chapter 9, p.158. When light strikes a highly polished surface, such as a mirror, it is reflected back into the same medium. This process is governed by two universal Laws of Reflection: first, the angle of incidence (the angle the incoming ray makes with a perpendicular line called the 'normal') is always equal to the angle of reflection; and second, the incident ray, the reflected ray, and the normal at the point of incidence all lie within the same plane Science, Class X (NCERT 2025 ed.), Chapter 9, p.135. These laws are absolute and apply to all reflecting surfaces, whether they are flat or curved Science, Class X (NCERT 2025 ed.), Chapter 9, p.158.

While we often think of plane mirrors and spherical mirrors as different entities, they are geometrically linked. A spherical mirror is essentially a slice of a hollow sphere. The radius of that original sphere is known as the Radius of Curvature (R), and the distance from the mirror to its focal point is the Focal Length (f). For these mirrors, a simple and elegant relationship exists: R = 2f Science, Class X (NCERT 2025 ed.), Chapter 9, p.137. This means the focus is exactly halfway between the mirror's surface and the center of the sphere it was cut from.

Using this logic, we can view a plane mirror as the extreme limit of a spherical mirror. Imagine a sphere getting larger and larger; as it grows, its surface becomes flatter. A perfectly flat plane mirror can be mathematically modeled as a sphere with an infinite radius of curvature Science, Class VIII, NCERT (Revised ed 2025), Chapter 10, p.160. Consequently, because R = 2f, the focal length of a plane mirror is also infinite. This is why a plane mirror does not converge or diverge light to a specific point—parallel rays simply remain parallel after reflection.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.134, 135, 137, 158; Science, Class VIII, NCERT (Revised ed 2025), Chapter 10: Light: Mirrors and Lenses, p.160

2. Properties of Images: Real, Virtual, and Lateral Inversion (basic)

To understand how we see objects in mirrors or through lenses, we must first distinguish between the two types of images light can create: Real and Virtual. A Real image is formed when light rays actually intersect at a point after reflection or refraction. Because the rays truly meet there, a real image can be captured on a screen (like a cinema screen). In contrast, a Virtual image is formed when light rays do not actually meet but only appear to diverge from a point behind the mirror or lens. You cannot project a virtual image onto a screen because the light rays never actually reach that point Science, Class VII, Chapter 11, p.161. There is a standard relationship between the nature of an image and its orientation. Generally, real images are inverted (upside down), while virtual images are erect (upright). This is why the image you see of yourself in a plane mirror is always upright—it is a virtual image. In mathematics and physics problems, we use signs to denote this: a positive magnification indicates a virtual and erect image, while a negative magnification indicates a real and inverted image Science, Class X, Chapter 9, p.143. Another fascinating property is Lateral Inversion. This is the phenomenon where the left side of the object appears as the right side of the image, and vice-versa. While many students think this only happens in plane mirrors, it is actually a property seen in all three types of mirrors: plane, concave, and convex Science, Class VIII, Chapter 10, p.156. This is the reason the word 'AMBULANCE' is written in reverse on the front of emergency vehicles—so that drivers ahead can read it correctly in their rear-view mirrors.

Feature	Real Image	Virtual Image
Ray Path	Rays actually meet/intersect.	Rays appear to meet when produced backwards.
Screen	Can be obtained on a screen.	Cannot be obtained on a screen.
Orientation	Inverted (with respect to object).	Erect (with respect to object).

Sources: Science, Class VII, Chapter 11: Shadows and Reflections, p.161; Science, Class VIII, Chapter 10: Light: Mirrors and Lenses, p.156; Science, Class X, Chapter 9: Light – Reflection and Refraction, p.143

3. Refraction and Total Internal Reflection (TIR) (intermediate)

When light travels from one transparent medium to another, it rarely continues in a perfectly straight line; instead, it bends at the interface. This phenomenon is known as Refraction. This bending occurs because light travels at different speeds in different materials. According to the laws of refraction, the incident ray, the refracted ray, and the normal at the point of incidence all lie in the same plane Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148. The most critical rule here is 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. This constant is called the refractive index of the second medium relative to the first.

As light moves from an optically denser medium (like water or glass) to an optically rarer medium (like air), it bends away from the normal. If you gradually increase the angle of incidence in the denser medium, the angle of refraction in the rarer medium also increases until it reaches 90°. At this specific point, the refracted ray grazes the surface of the interface. The angle of incidence that produces this 90° refraction is called the Critical Angle.

If the angle of incidence is increased even further beyond this critical angle, the light can no longer pass into the second medium. Instead, it is reflected entirely back into the denser medium. This fascinating phenomenon is called Total Internal Reflection (TIR). For TIR to occur, two conditions must be met:

• The light must be traveling from a denser medium to a rarer medium.
• The angle of incidence must be greater than the critical angle for that pair of media.

This principle is the reason why diamonds sparkle so brilliantly and how optical fibers carry high-speed internet data over vast distances.

In nature, these principles create stunning visual effects. For instance, rainbows and halos around the sun or moon are caused by a combination of reflection, refraction, and total internal reflection of sunlight by water droplets or ice crystals in the atmosphere Physical Geography by PMF IAS, Hydrological Cycle (Water Cycle), p.335. While refraction splits the light into its constituent colors (dispersion), TIR ensures the light is directed back toward the observer's eye, creating the circular arc we recognize.

Sources: Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148; Physical Geography by PMF IAS, Hydrological Cycle (Water Cycle), p.335

4. Optical Instruments and Power of Lenses (intermediate)

To understand optical instruments, we must first master the concept of the Power of a Lens. In simple terms, power represents the degree of convergence or divergence a lens can impose on light rays. A lens with a short focal length bends light more sharply, meaning it is more 'powerful' than one with a long focal length. Mathematically, the power (P) of a lens is the reciprocal of its focal length (f) in metres: P = 1/f. The SI unit of power is the dioptre (D), where 1D = 1m⁻¹ Science, Class X, Chapter 9: Light – Reflection and Refraction, p. 158.

The nature of the lens dictates the sign of its power. This is a critical distinction for both physics problems and practical optometry. As per the sign convention, a convex lens (converging) has a positive power, while a concave lens (diverging) has a negative power Science, Class X, Chapter 9: Light – Reflection and Refraction, p. 158. For instance, if an optician prescribes a lens of +2.0 D, you immediately know it is a convex lens with a focal length of +0.50 m.

Lens Type	Nature	Focal Length (f)	Power (P)
Convex	Converging	Positive (+)	Positive (+)
Concave	Diverging	Negative (-)	Negative (-)

In sophisticated optical instruments like cameras, microscopes, and telescopes, a single lens often produces blurred or distorted images (aberrations). To solve this, engineers use a combination of lenses in contact. The total power (P) of such a system is simply the algebraic sum of the individual powers: P = P₁ + P₂ + P₃... Science, Class X, Chapter 9: Light – Reflection and Refraction, p. 158. This additive property allows designers to precisely calibrate light to reveal hidden worlds—from tiny microbes to distant galaxies Science, Class VIII, Chapter: The Invisible Living World, p. 9.

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158; Science, Class VIII (NCERT 2025 ed.), The Invisible Living World: Beyond Our Naked Eye, p.9

5. Geometry of Curved Surfaces: Center and Radius of Curvature (intermediate)

To understand the geometry of curved mirrors, we must first visualize them as segments of a hollow sphere. Imagine a glass ball: if you cut a slice out of it and silver one side, you create a spherical mirror. The Center of Curvature (C) is the center of that original sphere. It is important to note that this point is not on the mirror's surface, but lies in front of a concave mirror or behind a convex mirror. The distance from this center to any point on the mirror's reflecting surface is known as the Radius of Curvature (R) Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p. 136.

Geometrically, a straight line passing through the center of curvature and the Pole (P)—the geometric center of the mirror's surface—is called the Principal Axis. This axis acts as a normal to the mirror at the pole. A fundamental relationship exists between the radius and the focal length: for mirrors with a small aperture (the diameter of the reflecting area), the radius of curvature is exactly twice the focal length (f), or R = 2f Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p. 137. This means the principal focus (F) always sits exactly midway between the pole and the center of curvature.

What happens when a mirror is perfectly flat? We can think of a plane mirror as a spherical mirror with an infinite radius of curvature. As a curved surface becomes flatter, the sphere it belongs to must become larger and larger. In the limit of a perfectly flat surface, the center of curvature moves infinitely far away. Consequently, because R is infinite, the focal length (f = R/2) of a plane mirror is also infinite, which is why it neither converges nor diverges parallel light rays to a finite point.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.136; Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.137; Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.159

6. The Plane Mirror as a Limit of Spherical Geometry (exam-level)

To understand the plane mirror through the lens of geometry, we must first look at its spherical cousins. A spherical mirror is essentially a piece cut from a hollow sphere. The center of this imaginary sphere is called the Centre of Curvature (C), and its radius is the Radius of Curvature (R) Science, Class X (NCERT 2025 ed.), Chapter 9, p. 136. In these mirrors, there is a fixed mathematical relationship where the focal length (f) is exactly half of the radius of curvature, expressed as R = 2f Science, Class X (NCERT 2025 ed.), Chapter 9, p. 137. This focal length determines how sharply the mirror converges or diverges light.

Now, imagine what happens as you make that imaginary sphere larger and larger. If you have a small sphere, the surface is highly curved. As the radius increases—say, from the size of a tennis ball to the size of the Earth—the surface appears increasingly flat to anyone standing on it. In the mathematical limit, when the radius of curvature becomes infinitely large (R = ∞), the surface loses all its curvature and becomes a plane mirror. Consequently, because f = R/2, the focal length of a plane mirror is also infinite (f = ∞).

This "infinite" focal length has profound physical implications. In a concave or convex mirror, parallel rays of light are forced to either meet at a point or appear to spread out from one. However, in a plane mirror, because the focus is at infinity, incoming parallel rays remain parallel after reflection. This is why a plane mirror produces an image that is the exact same size as the object, leading to a magnification of +1 Science, Class X (NCERT 2025 ed.), Chapter 9, p. 160. It is the simplest case of reflection where the "bending" of light rays is zero.

Feature	Spherical Mirror	Plane Mirror (The Limit)
Radius of Curvature (R)	Finite value	Infinite (∞)
Focal Length (f)	Finite (R/2)	Infinite (∞)
Curvature	Present	Zero

Sources: Science, class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.136; Science, class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.137; Science, class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.160

7. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental properties of light, you can see how the building blocks of spherical mirrors apply to this question. As established in Science, class X (NCERT 2025 ed.) > Chapter 9: Light – Reflection and Refraction, the radius of curvature (R) is the radius of the conceptual sphere from which a mirror is cut. To solve this, you must apply limiting logic: as a mirror's surface becomes less curved (flatter), the sphere it belongs to must become larger. Consequently, a perfectly flat plane mirror is mathematically treated as a spherical mirror with an infinite radius, meaning its focal length is also infinite because it cannot converge parallel rays to a finite point.

Walking through the reasoning, think of a large circle that gradually expands. As the circle grows, any small arc on its circumference looks increasingly like a straight line. In the limit where the line is perfectly straight—as in a plane mirror—the circle has grown so large that its center is at an infinite distance. This geometric interpretation ensures the laws of reflection remain consistent across all surfaces. Therefore, the correct answer is (B) is infinity, a direct application of the relationship where curvature is the inverse of the radius.

UPSC often uses specific traps to test your conceptual clarity. Option (A) is a common mistake where students confuse zero curvature (the state of being flat) with a zero radius; however, a radius of zero would physically represent a single point, not a plane. Option (C) is a distractor designed to tempt students who are unsure if "plane" refers to a specific geometric state or a general category. By sticking to the NCERT definition found in Science, Class VIII NCERT (Revised ed 2025) > Chapter 10: Light: Mirrors and Lenses, you can confidently identify that a flat surface always represents the infinite limit of a sphere.