A ray of light falls on a transparent glass plate. A part of it is reflected and a part is refracted. The reflected and refracted rays can be perpendicular to each other for

1. Fundamental Laws: Reflection and Refraction (basic)

Welcome to our journey into Geometrical Optics! To understand how the world around us is visible, we must first master the two fundamental ways light interacts with matter: Reflection and Refraction. Think of these as the 'Constitutional Rules' that light never breaks, whether it is hitting a bathroom mirror or passing through a glass of water.

The Laws of Reflection govern how light 'bounces' off a surface. There are two essential rules to remember: First, the angle of incidence (i) is always equal to the angle of reflection (r). Second, the incident ray, the reflected ray, and the 'normal' (an imaginary line perpendicular to the surface at the point of impact) all lie in the same plane Science, Class X, Light – Reflection and Refraction, p.135. A common misconception is that these laws only apply to flat mirrors; in reality, they apply to every reflecting surface, including curved ones like spoons or security mirrors Science, Class X, Light – Reflection and Refraction, p.158.

The Laws of Refraction describe how light 'bends' when it travels from one medium (like air) into another (like glass or water). This happens because light changes speed in different materials Science, Class X, Light – Reflection and Refraction, p.159. Similar to reflection, the incident ray, the refracted ray, and the normal all lie in the same plane. However, the unique rule here is Snell’s Law: the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for a given pair of media Science, Class X, Light – Reflection and Refraction, p.148. This constant is known as the Refractive Index (n), and it tells us how much the light will bend.

Sources: Science, Class X, Light – Reflection and Refraction, p.135; Science, Class X, Light – Reflection and Refraction, p.148; Science, Class X, Light – Reflection and Refraction, p.158; Science, Class X, Light – Reflection and Refraction, p.159

2. The Role of Refractive Index (n) (basic)

When light travels from one medium to another, it changes its speed, which causes it to bend. The Refractive Index (n) is the fundamental numerical value that describes this behavior. Think of it as a material's "optical resistance." Formally, the absolute refractive index (nₘ) of a medium is the ratio of the speed of light in a vacuum (c) to the speed of light in that specific medium (v). This is expressed as n = c/v Science Class X, Light – Reflection and Refraction, p.149. Because the speed of light is fastest in a vacuum, the refractive index for any material medium is always greater than 1.

It is crucial to distinguish between mass density and optical density. Mass density is simply mass per unit volume Science Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.140. However, optical density refers specifically to a medium's ability to slow down and refract light. A medium with a higher refractive index is called optically denser, and light travels slower in it compared to an optically rarer medium Science Class X, Light – Reflection and Refraction, p.149. Interestingly, these two types of density don't always align; for example, kerosene has a higher refractive index (1.44) than water (1.33), meaning it is optically denser, even though it is physically less dense and floats on water.

Beyond just bending light, the refractive index determines unique phenomena like Brewster’s Law. When light hits a surface at a specific angle (the polarizing angle), the reflected and refracted rays become perpendicular to each other. This specific angle depends entirely on the refractive index of the material, mathematically linked by the relation tan(i) = n. Thus, 'n' is not just a constant; it is the fingerprint of how a material interacts with electromagnetic waves.

Sources: Science Class X, Light – Reflection and Refraction, p.148; Science Class X, Light – Reflection and Refraction, p.149; Science Class X, Light – Reflection and Refraction, p.150; Science Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.140

3. Total Internal Reflection (TIR) and Critical Angle (intermediate)

When light travels from one transparent medium to another, it typically undergoes both reflection and refraction. However, a fascinating phenomenon occurs when light attempts to move from an optically denser medium (like glass or water) to an optically rarer medium (like air). According to the laws of refraction, or Snell's Law, the ray bends away from the normal as it enters the rarer medium Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148.

As we gradually increase the angle of incidence in the denser medium, the angle of refraction in the rarer medium also increases, bending further away from the normal. Eventually, we reach a specific point called the Critical Angle (i_c). At this precise angle of incidence, the refracted ray does not enter the second medium but instead travels along the interface (the boundary) between the two media, making an angle of refraction of 90°.

If the angle of incidence is increased even slightly beyond this critical angle, refraction can no longer occur. Instead, the boundary acts like a perfect mirror, and all the light is reflected back into the denser medium. This is known as Total Internal Reflection (TIR). Unlike ordinary mirrors, which absorb some light, TIR is 100% efficient, which is why it is used in high-precision technologies like fiber optics.

Sources: Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148; Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.134

4. Dispersion and Scattering of Light (intermediate)

When we see white light, we are actually looking at a combination of several colors. Dispersion is the phenomenon where this composite white light splits into its constituent colors (the VIBGYOR spectrum) when passing through a transparent medium like a glass prism. This happens because the refractive index of a material is not the same for all colors; it depends on the speed of light in that medium Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148. Different colors travel at different speeds in glass, causing them to bend at different angles. Violet light travels the slowest and thus bends the most, while red light travels the fastest and bends the least Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.167.

While dispersion is about the splitting of light due to refraction, Scattering is about the redirection of light. When light strikes small particles (like molecules in the atmosphere or dust), it is absorbed and then re-emitted in all directions. This is famously known as the Tyndall effect. The color of the scattered light depends on the size of the scattering particles. For example, very fine particles in the atmosphere scatter shorter wavelengths (blue) more effectively than longer wavelengths (red), which is why the clear sky appears blue.

Sources: Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.165-167; Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148

5. Wave Nature: Polarization of Light (intermediate)

To understand Polarization, we first need to look at the "shape" of a light wave. Unlike sound, which travels as a longitudinal wave (compressing and stretching the air), light is a transverse electromagnetic wave Physical Geography by PMF IAS, Earths Magnetic Field, p.64. This means the vibrations of the electric field occur perpendicular to the direction the light is traveling Physical Geography by PMF IAS, Earths Interior, p.62. In ordinary sunlight or light from a bulb, these vibrations happen in every possible plane—up-down, left-right, and diagonally. Polarization is the process of filtering these vibrations so they occur in only one single plane.

One of the most fascinating ways light becomes polarized is through reflection. When unpolarized light strikes a transparent surface, like a glass plate, the reflected light is usually partially polarized. However, there is a specific angle of incidence called Brewster’s Angle (or the polarizing angle) where the magic happens. At this precise angle, the reflected ray becomes completely polarized, with its vibrations parallel to the surface. For this to occur, a unique geometric condition must be met: the reflected ray and the refracted ray must be exactly 90° (perpendicular) to each other.

This relationship leads us to a simple mathematical rule known as Brewster’s Law. Since the angle of reflection equals the angle of incidence ($i$), and the sum of the angles at this specific point makes the rays perpendicular, we find that the refractive index ($n$) of the medium is equal to the tangent of the polarizing angle. This is expressed as: $tan(i) = n$. Because the refractive index is a constant for a specific material (like a particular type of glass), there is only one unique angle for any given material where this perfect perpendicularity—and thus total polarization by reflection—occurs.

Sources: Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.134; Physical Geography by PMF IAS, Earths Interior, p.62; Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64

6. Brewster's Law and the Polarizing Angle (exam-level)

When a beam of unpolarized light (light vibrating in all directions) strikes a transparent surface like glass or water, it typically splits into two: a reflected ray and a refracted ray. David Brewster discovered that at one specific angle of incidence, the reflected light becomes completely plane-polarized, meaning it vibrates in only one plane. This specific angle is known as Brewster’s angle or the polarizing angle (iₚ).

The physical hallmark of Brewster's Law is a unique geometric relationship: at the polarizing angle, the reflected ray and the refracted ray are exactly perpendicular (90°) to each other. To understand why this happens, we look at the fundamental laws of light. We know from the laws of reflection that the angle of incidence equals the angle of reflection Science, Light – Reflection and Refraction, p.135. At Brewster's angle, the sum of the angle of reflection and the angle of refraction is exactly 90°. This alignment causes the dipoles in the material to oscillate in a way that they cannot radiate energy in the direction of the reflected ray if that light were vibrating in the plane of incidence, leaving only the perpendicular vibrations to be reflected.

Mathematically, we can derive Brewster's Law using Snell’s Law, which states that the refractive index (n) is the ratio of the sine of the angle of incidence to the sine of the angle of refraction Science, Light – Reflection and Refraction, p.148. If the angle of incidence is iₚ and the angle of refraction is r, then iₚ + r = 90°, which means r = 90° - iₚ. Substituting this into Snell's Law: n = sin(iₚ) / sin(90° - iₚ). Since sin(90° - θ) = cos(θ), the formula simplifies to n = sin(iₚ) / cos(iₚ), or simply n = tan(iₚ). This tells us that for any given medium, there is only one specific angle where this perfect polarization occurs.

Sources: Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.135; Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148

7. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental laws of reflection and refraction, this question challenges you to apply them to a specific geometric constraint. In your previous lessons, you learned that light often splits at a boundary; here, we are looking for the exact moment that split creates a 90-degree angle between the two resulting rays. This phenomenon is governed by Brewster’s Law. By synthesizing Snell's Law with the law of reflection, we find that this perpendicular condition is only met when the tangent of the angle of incidence equals the refractive index of the glass ($tan(i) = n$). Because the refractive index is a constant physical property for a specific medium like a glass plate, the mathematical result is a single, unique value.

To arrive at the correct answer, (C) only one angle of incidence, you must visualize the relationship: as the angle of incidence changes, the refracted ray also shifts. There is only one specific point—the polarizing angle—where the geometry aligns perfectly to satisfy the perpendicularity requirement. UPSC often uses options like (D) to tempt students who might assume physical phenomena are variable or flexible, but the strict functional relationship between the angle and the refractive index ($n$) allows for no more than one solution in this context, as noted in Britannica.

It is equally important to recognize why the other options are classic "distractors." Option (B), angle of incidence equal to zero, is a common trap; at zero degrees (normal incidence), the rays are actually collinear (lying on the same line), not perpendicular. Option (A), 90° incidence, refers to grazing incidence where light skims the surface, failing to produce the necessary refracted path. By eliminating these extremes and trusting the uniqueness of the Brewster angle formula, you can confidently identify that this physical state occurs at only one specific instance.