A ray of light is incident normally on one of the faces of right angled isosceles prism as shown below. It undergoes total internal reflection from hypotenuse. Which one of the following is the minimum refractive index of the material of the prism?

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

Welcome to your first step in mastering Geometrical Optics! To understand how lenses, prisms, and even our own eyes work, we must first master the fundamental behavior of light as it crosses boundaries. This phenomenon is called Refraction.

Refraction occurs because light travels at different speeds in different materials. When a ray of light traveling through air hits a glass slab at an angle, it doesn't just keep going straight; it bends. This bending happens at the interface (the boundary) of the two media Science, Chapter 9, p.147. To predict which way the light will bend, we always draw an imaginary line perpendicular to the surface called the Normal.

To calculate exactly how much the light bends, we use Snell’s Law. It tells us that for a specific color of light and a specific pair of media, the ratio of the sine of the angle of incidence (i) to the sine of the angle of refraction (r) is a constant Science, Chapter 9, p.148. This constant is known as the Refractive Index (n) of the second medium with respect to the first. Mathematically, it is expressed as:

sin i / sin r = constant (n)

If light strikes the surface perfectly normally (at 90° to the surface, meaning the angle of incidence is 0°), it enters the second medium without any deviation in its path, though its speed still changes! This is a crucial detail to remember for complex optical instruments.

Sources: Science, Chapter 9: Light – Reflection and Refraction, p.147; Science, Chapter 9: Light – Reflection and Refraction, p.148

2. Understanding Refractive Index and Optical Density (basic)

Hello! To understand how light behaves in a prism or a lens, we first need to master the concept of the Refractive Index (n). Think of it as a measure of how much a medium "resists" the speed of light. When light travels from the vacuum of space into a material like glass or water, it interacts with the atoms and slows down. The refractive index is simply the ratio that tells us how much slower it goes. For example, if we say the absolute refractive index of a medium is 1.5, it means light travels 1.5 times faster in a vacuum than it does in that specific medium Science, Class X (NCERT 2025 ed.), Chapter 9, p.148.

Mathematically, we express this as n = c / v, where c is the speed of light in a vacuum (approximately 3 × 10⁸ m/s) and v is the speed of light in the medium. Because it is a ratio of two similar quantities, the refractive index has no units. It is a pure number. When comparing two different materials, we use the term Relative Refractive Index. If light travels from medium 1 to medium 2, the refractive index of medium 2 with respect to 1 (n₂₁) is the ratio of the speed of light in medium 1 to that in medium 2 Science, Class X (NCERT 2025 ed.), Chapter 9, p.148.

A common point of confusion in physics is the difference between Mass Density and Optical Density. Mass density is simply mass per unit volume (how heavy it is), but optical density refers specifically to how a medium affects the speed of light. Interestingly, a material can be mass-wise lighter but optically denser! For instance, Kerosene (n = 1.44) has a higher refractive index than Water (n = 1.33), meaning it is optically denser, even though kerosene floats on water because its mass density is lower Science, Class X (NCERT 2025 ed.), Chapter 9, p.149.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.148; Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.149; Science, Class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.150

3. Dispersion and the Electromagnetic Spectrum (intermediate)

When white light enters a triangular glass prism, it doesn't just bend; it beautifully unravels into a band of colors. This phenomenon is known as Dispersion. The root cause lies in a fundamental property of matter: the refractive index of a medium is not a single fixed number, but actually varies slightly depending on the wavelength (color) of the light passing through it. While all colors of light travel at the same speed in a vacuum, they travel at different speeds through glass. Science, class X (NCERT 2025 ed.), Chapter 10: The Human Eye and the Colourful World, p.167

In a prism, the two lateral faces are inclined at an angle called the angle of the prism. This geometry ensures that the different colors, which bend at different angles due to their unique speeds, emerge along distinct paths. Red light, having the longest visible wavelength, travels the fastest in glass and is deviated the least. Conversely, violet light has the shortest visible wavelength, travels the slowest, and is deviated the most. This separation creates the visible spectrum, famously first demonstrated by Isaac Newton. Science, class X (NCERT 2025 ed.), Chapter 10: The Human Eye and the Colourful World, p.166

Moving beyond what our eyes can see, the Electromagnetic (EM) Spectrum encompasses a vast range of waves, from high-energy Gamma rays to long-range Radio waves. The behavior of these waves depends heavily on their frequency. For instance, in our atmosphere, the ionosphere acts like a mirror for certain radio waves. If the frequency is below a "critical frequency," the free electrons in the ionosphere vibrate and re-radiate the energy back to Earth, allowing for long-distance communication. However, higher-frequency waves like microwaves carry more energy but are often absorbed or passed through the atmosphere rather than reflected. Physical Geography by PMF IAS, Earths Atmosphere, p.278-279

Sources: Science, class X (NCERT 2025 ed.), Chapter 10: The Human Eye and the Colourful World, p.165-167; Physical Geography by PMF IAS, Earths Atmosphere, p.278-279

4. Scattering and Atmospheric Optical Phenomena (intermediate)

When we look at the world around us, light doesn't always travel in perfectly straight lines. Often, it interacts with tiny particles in the atmosphere—a phenomenon we call scattering. Think of scattering as the deflection of light in various directions when it hits obstacles like gas molecules, dust, or water droplets. This interplay is responsible for some of nature's most beautiful spectacles, such as the blue sky and the crimson hues of a sunset Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.169.

The Tyndall Effect is a classic example of scattering that you can see in daily life. When a beam of sunlight enters a dusty room or passes through a dense forest canopy, the path of the light becomes visible because aerosols (like smoke and mist) scatter the light toward our eyes Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.169. Interestingly, the color we see depends heavily on the size of the scattering particles. Very fine particles, like nitrogen or oxygen molecules in the air, are better at scattering shorter wavelengths (blue and violet). Larger particles, like water droplets in clouds, scatter all wavelengths of light almost equally, which is why clouds appear white Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.169.

For your UPSC preparation, it is vital to understand the relationship between wavelength and particle size. If the wavelength of the light is greater than the radius of the particle (like a gas molecule), scattering occurs. However, if the particle is much larger (like a coarse dust particle), the light might simply reflect off it rather than scatter Physical Geography by PMF IAS, Horizontal Distribution of Temperature, p.283. This explains why the sky is blue during the day: the sun is overhead, and the fine gas molecules scatter the blue light most effectively. At sunrise or sunset, the light has to travel a much thicker layer of the atmosphere; the shorter blue wavelengths are scattered away long before reaching us, leaving only the longer red wavelengths to reach our eyes.

Sources: Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.169; Physical Geography by PMF IAS, Horizontal Distribution of Temperature, p.283

5. Optical Instruments and Corrective Lenses (intermediate)

Optical instruments like the human eye work by focusing light onto a specific plane. In a healthy eye, the crystalline lens adjusts its focal length to form a sharp image on the retina. However, when the eye's power of accommodation fails, refractive defects occur. These are primarily Myopia (near-sightedness), where the image forms in front of the retina, and Hypermetropia (far-sightedness), where the image forms behind it Science, Class X, The Human Eye and the Colourful World, p.162. These defects are corrected using spherical lenses that shift the image back onto the retinal surface.

To quantify how much a lens can bend light, we use the Power of a Lens (P). Power is mathematically the reciprocal of the focal length (f) measured in meters (P = 1/f). The SI unit is the dioptre (D). By convention, a convex lens (converging) has positive power, while a concave lens (diverging) has negative power Science, Class X, Light – Reflection and Refraction, p.158. For example, a lens with a power of +2.0 D is a convex lens with a focal length of +0.5 meters.

In addition to eye correction, optical instruments like prisms utilize Total Internal Reflection (TIR). For a right-angled isosceles prism to act as a perfect reflector (bending light by 90° or 180°), the light must strike the internal surface at an angle greater than the critical angle (θc). Since the geometry of such a prism dictates an incidence angle of 45°, the refractive index (n) must satisfy the condition n ≥ 1/sin 45°. This means the prism material must have a minimum refractive index of √2 (approx. 1.414) to ensure light does not leak out Science, Class X, Light – Reflection and Refraction, p.148.

Sources: Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.162-164; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148

6. Total Internal Reflection (TIR) and Critical Angle (exam-level)

When light travels from an optically denser medium (like glass or water) to an optically rarer medium (like air), it bends away from the normal Science, Light – Reflection and Refraction, p.147. As we increase the angle of incidence (i), the angle of refraction (r) also increases. Eventually, we reach a specific point called the Critical Angle (θc). At this precise angle, the refracted ray doesn't escape into the air but instead grazes along the boundary, making the angle of refraction exactly 90°.

If the incident angle is increased even slightly beyond this critical angle, refraction becomes impossible. Instead, the boundary acts like a perfect mirror, and the light is reflected entirely back into the denser medium. This phenomenon is known as Total Internal Reflection (TIR). Unlike reflection from a silvered mirror, which absorbs some light, TIR is "total" because 100% of the light energy is retained within the medium, which is why it is used in high-precision instruments like fiber optics and periscopes Science, Light – Reflection and Refraction, p.134.

To calculate the threshold for TIR, we use Snell's Law, which states that the ratio of the sines of the angles of incidence and refraction is a constant Science, Light – Reflection and Refraction, p.148. For light exiting a medium of refractive index n into air (where refractive index is ~1), the relationship is defined as:

sin θc = 1/n

Consider a right-angled isosceles prism (angles: 45°, 90°, 45°). If a ray enters one face normally, it strikes the internal hypotenuse at an angle of 45°. For this ray to undergo TIR, the critical angle of the glass must be 45° or less. Using our formula (sin 45° ≥ 1/n), we find that n must be at least √2 (approx. 1.414). If the glass has a lower refractive index, the light will simply refract out and escape the prism.

Sources: Science, Light – Reflection and Refraction, p.147; Science, Light – Reflection and Refraction, p.134; Science, Light – Reflection and Refraction, p.148

7. Ray Tracing in Right-Angled Isosceles Prisms (exam-level)

In geometrical optics, the right-angled isosceles prism (with angles of 90°, 45°, and 45°) is a precision tool used to manipulate the path of light through Total Internal Reflection (TIR). Unlike a standard triangular prism where light usually refracts through the faces Science, The Human Eye and the Colourful World, p.166, these prisms are often designed to act as perfect mirrors.

When a light ray strikes one of the shorter faces (the legs) of the prism at a normal incidence (90° to the surface), it enters the glass without any deviation Science, Light – Reflection and Refraction, p.147. Because of the prism's geometry, this ray travels straight until it hits the hypotenuse. At this interface, the ray meets the surface at an angle of incidence (θ) of 45°. Whether the light exits the prism or reflects back depends entirely on the material's refractive index (n).

For the ray to undergo TIR rather than refracting out into the air, the angle of incidence must be greater than or equal to the critical angle (θc). We determine this using the relationship sin θc = 1/n. Since our angle of incidence is fixed by the prism's shape at 45°, we require:

• sin 45° ≥ 1/n
• Since sin 45° = 1/√2, the condition becomes 1/√2 ≥ 1/n
• Rearranging this, we find n ≥ √2 (which is approximately 1.414).

If the prism is made of standard glass (where n ≈ 1.5), the critical angle is roughly 42°. Since 45° is greater than 42°, the light reflects perfectly. This principle allows these prisms to deviate light by 90° or even 180° (if the light enters through the hypotenuse), making them essential in devices like periscopes and binoculars.

Sources: Science, The Human Eye and the Colourful World, p.166; Science, Light – Reflection and Refraction, p.147

8. Solving the Original PYQ (exam-level)

This question is a brilliant synthesis of the building blocks you have just mastered: Snell’s Law, the principle of normal incidence, and the condition for Total Internal Reflection (TIR). When the ray enters the first face of the prism normally, it travels undeviated because the angle of incidence is zero. By applying the basic geometry of a 45°-90°-45° triangle, we can deduce that the ray strikes the internal hypotenuse at an angle of incidence of exactly 45°. This sets the stage for the final step: determining if the light will reflect or refract.

To find the minimum refractive index (n), we look for the "threshold" where the incident angle is exactly equal to the critical angle. According to the TIR formula, $sin(i_c) = 1/n$. Since our angle is 45°, we set $sin(45°) = 1/n$. Recalling your trigonometric values, $sin(45°)$ is $1/\sqrt{2}$. By rearranging the equation ($1/\sqrt{2} = 1/n$), we find that $n = \sqrt{2}$. Calculating the square root of 2 gives us approximately 1.414. As explained in Science, Class X (NCERT), any medium with a refractive index lower than this would allow the light to escape the prism, while any value equal to or higher will satisfy the condition for reflection.

UPSC often includes distractors to test your conceptual clarity. Option (B) 1.33 is the refractive index of water, which is a common value but numerically insufficient to cause TIR at this specific angle. Option (D) 1.6 is a common value for dense glass; while TIR would occur at 1.6, the question specifically asks for the minimum value. Always be wary of the word "minimum" in physics problems—it is a cue to solve for the exact mathematical boundary (the limit) rather than just any value that works. Therefore, 1.414 is the only correct threshold value.