Light: Refractive Index, Snell's Law & Glass Slab Class 10

When a straight wooden pencil is placed in a glass tumbler filled with water, it appears strangely broken or bent at the liquid surface. When a thick glass slab is laid over a printed book, the letters appear magically lifted towards the viewer. These everyday optical illusions are governed by the bending of light as it transitions between transparent optical media of differing optical densities: the phenomenon of Refraction of Light.

In CBSE Class 10 Science, Chapter 9 (Light - Reflection and Refraction), understanding Snell's Law, calculating Absolute and Relative Refractive Indices, and tracing refraction through a Rectangular Glass Slab represent high-frequency questions in Section B and Section C.

In this master guide, we break down the mathematical formulas, optical principles, and solved board exam numericals.

What You Will Learn

• Why does light bend? The wave-speed mechanism of refraction
• The two fundamental Laws of Refraction of Light
• Snell's Law: $\frac{\sin i}{\sin r} = \text{constant} = n_{21}$
• Absolute Refractive Index ($n = c/v$) vs. Relative Refractive Index ($n_{21} = v_1/v_2 = n_2/n_1$)
• Optical Density vs. Mass Density (Why kerosene floats on water yet is optically denser!)
• Refraction through a Rectangular Glass Slab and proving $\angle i = \angle e$
• Factors governing Lateral Displacement ($d$)
• Solved CBSE board examination numerical problems

1. The Physical Cause of Refraction

Light travels in a vacuum at the cosmic speed limit: $c = 3 \times 10^8\text{ m/s}$. However, when a light beam enters an optically transparent medium (water, glass, diamond), its speed decreases due to interactions with the medium's atomic electrons.

    Case 1: Rarer to Denser Medium               Case 2: Denser to Rarer Medium
    (Speed DECREASES!)                            (Speed INCREASES!)
            Incident Ray                                  Incident Ray
                 \                                                               \ ∠i                                          \ ∠i
    ~~~~~~~~~~~~~~~+~~~~~~~~~~~~~~~               ~~~~~~~~~~~~~~~+~~~~~~~~~~~~~~~
                   | \                                           |                      |  \ ∠r                                       |     \ ∠r
    Bends TOWARDS the Normal! (∠i > ∠r)           Bends AWAY from the Normal! (∠i < ∠r)

2. The Laws of Refraction and Snell's Law

1. First Law: The incident ray, the refracted ray, and the normal to the interface of two transparent media at the point of incidence all lie in the same plane.
2. Second Law (Snell's Law of Refraction):

   The ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for the light of a given color and for a given pair of media: $\mathbf{\frac{\sin i}{\sin r} = \text{constant} = n_{21}}$

3. Absolute vs. Relative Refractive Index

A. Absolute Refractive Index ($n_m$):

When medium 1 is a vacuum (or air), the refractive index of medium 2 is called its Absolute Refractive Index: $\mathbf{n = \frac{\text{Speed of light in vacuum } (c)}{\text{Speed of light in medium } (v)} = \frac{c}{v}}$

• Since $c > v$ in all transparent materials, the absolute refractive index is always greater than or equal to $1$ ($n \ge 1$).
• Examples: Air $\approx 1.0003$, Water $= 1.33 = \frac{4}{3}$, Crown Glass $= 1.52$, Diamond $= 2.42$ (highest optical density!).

B. Relative Refractive Index ($n_{21}$):

The refractive index of medium 2 with respect to medium 1: $\mathbf{n_{21} = \frac{\text{Speed of light in medium 1 } (v_1)}{\text{Speed of light in medium 2 } (v_2)} = \frac{n_2}{n_1}}$

• Similarly: $n_{12} = \frac{1}{n_{21}}$ (Reciprocal property!).

Important: <u>Optical Density is NOT the same as Mass Density! Mass density is mass per unit volume ($\text{kg/m}^3$). Optical density is the degree to which a medium retards the speed of light. For example, KEROSENE has a lower mass density than water (it floats on water), but kerosene has a HIGHER refractive index ($n = 1.44$) than water ($n = 1.33$), making kerosene OPTICALLY DENSER than water!</u>

4. Refraction Through a Rectangular Glass Slab

When a ray of light passes through a rectangular glass slab with parallel refracting faces:

                            Incident Ray
                                                                 \ ∠i
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Interface 1 (Air -> Glass)
                                  | \ ∠r₁
                                  |  \ Refracted Ray inside glass
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+----+~~~~~~~~~~~~~~~~~~~~~~~~~~ Interface 2 (Glass -> Air)
                                  |     \ ∠e
                                  |      \ Emergent Ray (PARALLEL to incident ray!)
                     <-- Lateral Displacement d -->

Key Mathematical Inferences:

1. At Interface 1 (Air $\to$ Glass): $\frac{\sin i}{\sin r_1} = n_{\text{glass}}$.
2. At Interface 2 (Glass $\to$ Air): $\frac{\sin r_2}{\sin e} = \frac{1}{n_{\text{glass}}}$.
3. Because the refracting faces are parallel, alternate interior angles are equal: $r_1 = r_2$.
4. Equating the expressions establishes: $\mathbf{\sin i = \sin e \implies \angle i = \angle e}$

   <u>The angle of incidence is strictly equal to the angle of emergence! The emergent ray is parallel to the incident ray direction.</u>

What is Lateral Displacement ($d$)?

The perpendicular distance between the original path of the incident ray and the emergent ray is called Lateral Displacement ($d$).

• Factors Increasing Lateral Displacement ($d$):
  1. Directly proportional to the thickness ($t$) of the glass slab.
  2. Directly proportional to the angle of incidence ($i$).
  3. Directly proportional to the refractive index ($n$) of the glass.
  4. Inversely proportional to the wavelength ($\lambda$) of light (violet light undergoes more lateral shift than red light).

5. Solved Board Examination Problems

Solved Problem 1: Calculating Speed of Light in Glass

Problem: The absolute refractive index of glass is $1.50$. If the speed of light in a vacuum is $3 \times 10^8\text{ m/s}$, calculate the speed of light in glass.

Solution:

1. Formula: $n = \frac{c}{v} \implies v = \frac{c}{n}$.
2. Substitute values: $v = \frac{3 \times 10^8\text{ m/s}}{1.50} = \mathbf{2 \times 10^8\text{ m/s}}$
3. Therefore, <u>the speed of light in glass is $2 imes 10^8 ext{ m/s}$</u>.

Solved Problem 2: Relative Refractive Index

Problem: Light travels from water (refractive index $4/3$) into crown glass (refractive index $3/2$). Calculate the refractive index of glass with respect to water.

Solution:

1. Given: $n_w = \frac{4}{3}$ and $n_g = \frac{3}{2}$.
2. Refractive index of glass with respect to water ($n_{gw}$): $n_{gw} = \frac{n_g}{n_w} = \frac{\frac{3}{2}}{\frac{4}{3}} = \frac{3}{2} \times \frac{3}{4} = \mathbf{\frac{9}{8} = 1.125}$
3. Therefore, <u>the refractive index of glass with respect to water is rac{9}{8} (or $1.125$)</u>.

6. Summary and Examination Tips

Exam Tip: In ray diagrams of a glass slab, ALWAYS draw the forward dashed extension of the incident ray to clearly illustrate the lateral displacement ($d$)! Label $\angle i$, $\angle r$, $\angle e$, and the normal lines clearly.

Common Mistake: Confusing $n_{21}$ with $n_{12}$. $n_{21}$ is the refractive index of medium 2 with respect to medium 1, which equals $v_1/v_2$ (or $n_2/n_1$), NOT $v_2/v_1$!