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Refractive Index in Geometrical Optics
The refractive index is a dimensionless scalar quantity that quantifies the reduction of the speed of light within a transparent medium relative to its speed in a vacuum. It dictates the bending of light rays at optical interfaces in accordance with Snell's law.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Snell's law, absolute and relative refractive index, principle of reversibility of light, total internal reflection, and critical angle calculations for simple optical systems.
PGT extension
Wavelength dependence (Cauchy's dispersion relation), complex refractive index for absorbing media, group refractive index, and electromagnetic derivation relating refractive index to permittivity and permeability.
Prerequisites
Wave propagation and electromagnetic wave nature
Huygens' principle for wavefront construction
Basic trigonometry and vector geometry
Absolute and Relative Refractive Index
Absolute refractive index n of a medium is defined as the ratio of the speed of light in vacuum (c) to the speed of light in that medium (v), expressed as $n = c/v$.
Since c > v for all material media, the absolute refractive index is always greater than unity (n $ \rangle= 1)$. For vacuum, n = 1 exactly.
Relative refractive index n21 of medium 2 with respect to medium 1 is the ratio of their absolute refractive indices, given by n21 = n2 / n1 = v1 / v2.
Snell's Law of Refraction
When a light ray passes from an optically rarer medium (n1) to a denser medium (n2), it bends towards the normal.
Snell's law states that the product of the refractive index and the sine of the angle of incidence in one medium is equal to the corresponding product in the second medium: n1 sin(θ1) = n2 sin(θ2).
The incident ray, the refracted ray, and the normal to the interface at the point of incidence all lie in the same plane.
Total Internal Reflection (TIR) and Critical Angle
TIR occurs when light travels from an optically denser medium (n1) to a rarer medium (n2), where n1 > n2.
The critical angle (θc) is the angle of incidence in the denser medium for which the angle of refraction in the rarer medium is exactly 90 degrees.
Calculated using sin(θc) = n2 / n1. For angles of incidence greater than θc, the light reflects entirely back into the denser medium.
Electromagnetic and Dispersion Relations
From Maxwell's equations, the refractive index of a non-magnetic dielectric medium is related to its relative permittivity (εr) and relative permeability (μr) by $n = \sqrt(\epsilon_{r} \mu_{r})$. For most optical media, μr ≈ 1, so n ≈ sqrt(εr).
Cauchy's dispersion formula expresses the variation of refractive index with wavelength (λ) as n(λ) = A + B/λ2 + C/λ4, showing that shorter wavelengths experience higher refractive indices.
Group refractive index (ng) accounts for wave packet propagation and is defined as ng = c / vg = n - λ(dn/dλ).
Derivations
Derivation of Snell's Law using Huygens' Principle
Conditions: Plane wavefront incident on a plane interface separating two isotropic media.
Let a plane wavefront AB be incident at an angle i on the interface XY separating medium 1 (speed v1) and medium 2 (speed v2).
In time t, point B travels to B' covering distance v1t, while the secondary wavelet from A travels a distance v2t to reach point A' in medium 2, forming the refracted wavefront A'B'.
Relation between Refractive Index and Dielectric Constant
Conditions: Isotropic, homogeneous, non-magnetic dielectric medium (μr = 1).
The speed of an electromagnetic wave in a vacuum is given by $c = 1 / \sqrt(\mu_{0} \epsilon_{0})$.
The speed of an electromagnetic wave in a material medium is given by $v = 1 / \sqrt(\mu \epsilon) = 1 / \sqrt(\mu_{0} \mu_{r} \epsilon_{0} \epsilon_{r})$.
By definition, absolute refractive index $n = c / v$.
Substituting the expressions for c and v, $n = [1 / \sqrt(\mu_{0} \epsilon_{0})] / [1 / \sqrt(\mu_{0} \mu_{r} \epsilon_{0} \epsilon_{r})] = \sqrt(\mu_{r} \epsilon_{r})$.
For non-magnetic optical media, relative permeability μr ≈ 1, reducing the relation to $n = \sqrt(\epsilon_{r})$.
Result: $n = \sqrt(\epsilon_{r})$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Absolute Refractive Index
$$n = \frac{c}{v}$$
n = refractive index, c = speed of light in vacuum, v = speed of light in medium Dimensionless
Valid for monochromatic light in uniform media
Relative Refractive Index
$$n_{21} = \frac{n_2}{n_1} = \frac{v_1}{v_2}$$
$n_{21} = refractive index of medium 2 with respect$ to 1, v_1, $v_2 = speeds in media 1$ and 2 Dimensionless
Interface between two isotropic media
Snell's Law
$$n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$$
\thet$a_1 = angle of incidence$, \thet$a_2 = angle of refraction$ Radians or degrees
Specular interface between two transparent media
Critical Angle for Total Internal Reflection
$$\sin(\theta_c) = \frac{n_2}{n_1}$$
\thet$a_c = critical angle$, $n_1 = denser medium index$, $n_2 = rarer medium index$ Radians or degrees
n_1 > n_2
Cauchy's Dispersion Formula
$$n(\lambda) = A + \frac{B}{\lambda^2} + \frac{C}{\lambda^4}$$
\lambda = wavelength of light, A, B, $C = material-dependent dispersion constants$ A is dimensionless, B in m^2, C in m^4
Valid in spectral regions away from absorption bands
Group Refractive Index
$$n_g = n - \lambda \left(\frac{dn}{d\lambda}\right)$$
$n_g = group refractive index$, n = phase refractive index, \lambda = wavelength Dimensionless
Draw: A straight-line graph plot of sin(i) on the y-axis versus sin(r) on the x-axis passing through the origin.
Interpret: The slope of the straight line directly yields the relative refractive index n21 = n2/n1.
Exam use: Tested conceptually in graphical MCQ format where students identify slope parameters or intercept conditions.
Cauchy Dispersion Curve
Draw: A decreasing monotonic curve plotting refractive index n on the vertical axis against wavelength λ on the horizontal axis.
Interpret: Shows that refractive index decreases as wavelength increases (violet light bends more than red light).
Exam use: Directly tested in PGT questions concerning chromatic aberration and wavelength dependence.
Total Internal Reflection Tank Experiment
Draw: A light ray passing from water into air, showing critical refraction at θc and total reflection beyond θc.
Interpret: Demonstrates the cutoff angle where refracted power drops to zero and reflection coefficient becomes unity.
Exam use: Basis for numerical problems on optical fibers, prisms, and mirage phenomena.
Solved numericals
TGT core
Light enters from air into a glass slab of refractive index 1.5. If the angle of incidence in air is 45 degrees, calculate the angle of refraction in the glass. (Given sin(45°) = 0.707)
Identify given values: n1 (air) = 1.0, n2 (glass) = 1.5, i = 45 degrees.
The refractive index of a transparent medium varies with wavelength according to Cauchy's relation n(λ) = 1.20 + 0.50 * 10-14 / λ2, where λ is in meters. Calculate the group refractive index ng for light of wavelength 500 nm.
Given expression: n(λ) = A + B / λ2, where A = 1.20 and $B = 0.50 * 10^{-14} m^{2}$.
Total Internal Reflection occurs only from denser ($n_1$) to rarer ($n_2$) medium when angle of incidence exceeds critical angle $\theta_c = \sin^{-1}(n_2/n_1)$.
Frequency of light remains constant across medium boundaries, while speed and wavelength change.
Cauchy's dispersion formula: $n(\lambda) = A + \frac{B}{\lambda^2} + \frac{C}{\lambda^4}$.
Group refractive index accounts for wave packets: $n_g = n - \lambda \left(\frac{dn}{d\lambda}\right)$.
Electromagnetic relation: $n = \sqrt{\mu_r \varepsilon_r}$, with $\mu_r \approx 1$ for non-magnetic media giving $n \approx \sqrt{\varepsilon_r}$.
Glossary
Absolute Refractive Index ($n$)The ratio of the speed of light in vacuum to the speed of light in a given medium.
Relative Refractive Index ($n_{21}$)The ratio of the refractive index of medium 2 to that of medium 1, representing the ratio of wave speeds $v_1 / v_2$.
Critical Angle ($\theta_c$)The angle of incidence in the denser medium for which the angle of refraction in the rarer medium is $90^\circ$.
Cauchy's Constants ($A, B, C$)Empirical material-dependent constants used in Cauchy's dispersion relation to describe wavelength dependence of refractive index.
Group Refractive Index ($n_g$)Refractive index relevant to the propagation velocity of a wave packet or pulse in a dispersive medium.
Phase Refractive Index vs Group Refractive Index
Parameter
Phase Refractive Index ($n$)
Group Refractive Index ($n_g$)
Definition
$c / v_{phase}$
$c / v_{group}$
Equation
$n = \frac{c}{v}$
$n_g = n - \lambda \left(\frac{dn}{d\lambda}\right)$
Physical Meaning
Speed of phase fronts (crests/troughs)
Speed of energy and information transmission (wave packet)
Practice Quiz
Choose an option to reveal the explanation.
1. What is the absolute refractive index of a medium in which the speed of light is $2 \times 10^8 \text{ m/s}$?
Explanation: Using the absolute refractive index formula $n = \frac{c}{v}$, where $c = 3 \times 10^8 \text{ m/s}$ and $v = 2 \times 10^8 \text{ m/s}$, we get $n = \frac{3 \times 10^8}{2 \times 10^8} = 1.5$.
2. When a light ray passes from a rarer medium into a denser medium, how does it behave?
Explanation: According to Snell's law, when light travels from an optically rarer medium ($n_1$) to a denser medium ($n_2$), $n_2 > n_1$, causing the ray to bend towards the normal ($\theta_2 < \theta_1$).
3. Which of the following properties of light remains unchanged when it crosses the boundary from air into glass?
Explanation: Frequency is a fundamental property determined strictly by the source of light and remains invariant across optical boundaries to maintain boundary continuity.
4. What is the condition for Total Internal Reflection (TIR) to occur?
Explanation: TIR occurs exclusively when light travels from a denser medium to a rarer medium and the angle of incidence exceeds the critical angle ($\theta > \theta_c$).
5. If the critical angle for a medium-air interface is $30^\circ$, what is the refractive index of the medium?
Explanation: Using the critical angle formula $\sin(\theta_c) = \frac{n_2}{n_1}$, with $n_2 = 1$ (air) and $\theta_c = 30^\circ$, we have $\sin(30^\circ) = \frac{1}{n_1} \implies \frac{1}{2} = \frac{1}{n_1} \implies n_1 = 2.0$.
6. According to Cauchy's dispersion formula $n(\lambda) = A + \frac{B}{\lambda^2} + \frac{C}{\lambda^4}$, how does refractive index vary with wavelength?
Explanation: Cauchy's dispersion formula shows an inverse relationship between $n$ and $\lambda$ (via $\lambda^2$ and $\lambda^4$ terms in the denominator), meaning shorter wavelengths experience a higher refractive index.
7. What is the electromagnetic relation between refractive index $n$, relative permittivity $\varepsilon_r$, and relative permeability $\mu_r$ for a homogeneous linear dielectric medium?
Explanation: Derived from Maxwell's equations, the phase refractive index of a non-conducting medium is given by $n = \sqrt{\mu_r \varepsilon_r}$.
8. What is the group refractive index $n_g$ formula in terms of phase refractive index $n$ and wavelength $\lambda$?
Explanation: The group refractive index accounting for wave packet propagation in dispersive media is defined as $n_g = c / v_g = n - \lambda (dn/d\lambda)$.
9. For a non-magnetic material medium with relative permittivity $\varepsilon_r = 4$, what is its approximate absolute refractive index?
Explanation: For non-magnetic media, relative permeability $\mu_r \approx 1$. Using $n = \sqrt{\mu_r \varepsilon_r} \approx \sqrt{\varepsilon_r} = \sqrt{4} = 2.0$.
10. Which of the following statements about absolute refractive index is incorrect?
Explanation: The phase velocity of light in any passive material medium never exceeds $c$, meaning absolute refractive index $n = c/v$ can never be less than 1 under normal physical conditions.
11. If medium 1 has absolute refractive index $n_1$ and medium 2 has $n_2$, what is the relative refractive index $n_{12}$ of medium 1 with respect to medium 2?
Explanation: The relative refractive index of medium 1 with respect to medium 2 is defined as $n_{12} = \frac{n_1}{n_2} = \frac{v_2}{v_1}$.
12. In Snell's law $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$, what does the product $n \sin(\theta)$ represent across a boundary?
Explanation: Snell's law is fundamentally the conservation of the tangential component of the wave vector ($k \sin\theta$) across the flat interface.
13. Why do violet rays suffer greater deviation than red rays when passing through a prism?
Explanation: According to Cauchy's formula, shorter wavelengths (violet) have higher refractive indices than longer wavelengths (red), leading to greater refraction and deviation.
14. What happens to the wavelength of monochromatic light when it enters from air into a medium of refractive index $n$?
Explanation: Since frequency $f$ remains constant and $v = f \lambda$, with $v = c/n$, the wavelength in the medium becomes $\lambda_m = \frac{c}{f \cdot n} = \frac{\lambda_0}{n}$.
15. What is the Brewster angle ($i_p$) condition when light is incident on a transparent dielectric medium of refractive index $n$?
Explanation: Brewster's law states that at the polarizing angle, $\tan(i_p) = \frac{n_2}{n_1} = n$ when incident from air ($n_1 = 1$).
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The refractive indices of water and glass with respect to air are $4/3$ and $3/2$ respectively. What is the refractive index of glass with respect to water?
Explanation: The refractive index of glass with respect to water is $n_{gw} = \frac{n_g}{n_w} = \frac{3/2}{4/3} = \frac{3}{2} \times \frac{3}{4} = \frac{9}{8}$.
2. A ray of light travels from a denser medium into a rarer medium. If the critical angle for the media pair is $45^\circ$, what is the relative refractive index of the denser medium with respect to the rarer medium?
Explanation: Using $\sin(\theta_c) = \frac{n_2}{n_1}$, where $\theta_c = 45^\circ$ and $n_{21} = \frac{n_2}{n_1} = \sin(45^\circ) = \frac{1}{\sqrt{2}}$, the relative refractive index of denser with respect to rarer is $n_{12} = \frac{n_1}{n_2} = \sqrt{2}$.
3. If Cauchy's dispersion constants for a given optical glass are $A = 1.50$ and $B = 6.0 \times 10^{-14} \text{ m}^2$, what is the approximate refractive index for light of wavelength $\lambda = 400 \text{ nm}$ ($4 \times 10^{-7} \text{ m}$)?
Explanation: Using $n(\lambda) = A + \frac{B}{\lambda^2}$, we get $n = 1.50 + \frac{6.0 \times 10^{-14}}{(4 \times 10^{-7})^2} = 1.50 + \frac{6.0 \times 10^{-14}}{1.6 \times 10^{-13}} = 1.50 + 0.375 = 1.875$? Wait, let's recalculate: $6 \times 10^{-14} / 1.6 \times 10^{-13} = 6 / 1.6 = 3.75$? No, $(4 \times 10^{-7})^2 = 1.6 \times 10^{-13}$. $6 \times 10^{-14} = 0.6 \times 10^{-13}$. Thus $0.6 / 1.6 = 3/8 = 0.375$. Wait, $1.50 + 0.375 = 1.875$. Let's check options. Option D is $1.875$. Therefore, correct index is 3.
4. Light of frequency $5 \times 10^{14} \text{ Hz}$ passes through a medium of refractive index $1.5$. What is the wavelength of light in this medium?
5. What is the group refractive index $n_g$ if the phase refractive index $n$ satisfies $n = A + \frac{B}{\lambda^2}$ according to Cauchy's formula?
Explanation: Using $n_g = n - \lambda \frac{dn}{d\lambda}$. Since $\frac{dn}{d\lambda} = -\frac{2B}{\lambda^3}$, we get $n_g = \left(A + \frac{B}{\lambda^2}\right) - \lambda \left(-\frac{2B}{\lambda^3}\right) = A + \frac{B}{\lambda^2} + \frac{2B}{\lambda^2} = A + \frac{3B}{\lambda^2}$.
6. A transparent slab of thickness $t$ and refractive index $n$ is placed in the path of a light beam. What is the optical path length of the slab?
Explanation: Optical path length is defined as the product of the geometrical path length and the refractive index of the medium, i.e., $n t$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. The absolute refractive index of diamond is approximately $2.42$. What is the speed of light in diamond?
2. A ray of light is incident normally on a glass-air interface from inside glass. What is its angle of refraction in air?
Explanation: Normal incidence implies angle of incidence $\theta_1 = 0^\circ$. By Snell's law, $n_1 \sin(0) = n_2 \sin(\theta_2) \implies \theta_2 = 0^\circ$ with no deviation.
3. What is the critical angle for a medium with absolute refractive index $\sqrt{2}$ when surrounded by air?
5. Which of the following phenomena clearly demonstrates the wave nature of light and dispersion of refractive index?
Explanation: Rainbow formation is a direct consequence of refraction, total internal reflection, and wavelength-dependent dispersion of refractive index in water droplets.
6. The refractive index $n$ of a medium as a function of frequency $f$ typically exhibits normal dispersion where:
Explanation: Since refractive index decreases with wavelength ($\frac{dn}{d\lambda} < 0$) and frequency is inversely proportional to wavelength (\lambda = c/f), refractive index increases with frequency in normal dispersion regions ($\frac{dn}{df} > 0$).
7. A beam of light passes from medium A to medium B. If the angle of incidence is $45^\circ$ and the angle of refraction is $30^\circ$, what is the relative refractive index of medium B with respect to medium A ($n_{BA}$)?
8. In optical fiber communication, light is guided through the core due to:
Explanation: Optical fibers rely on continuous total internal reflection at the core-cladding boundary, where the core has a higher refractive index than the cladding.
9. What is the physical significance of a complex refractive index $\tilde{n} = n + i\kappa$ in absorbing media?
Explanation: In absorbing media, the real part $n$ corresponds to phase velocity and refraction, while the extinction coefficient $\kappa$ (imaginary part) accounts for spatial attenuation (absorption of light).
10. If the refractive index of a medium is given by $n = 2 - 0.5\lambda$, what can be inferred about group refractive index $n_g$ relative to $n$?
Explanation: Here $\frac{dn}{d\lambda} = -0.5$. Using $n_g = n - \lambda(dn/d\lambda) = n - \lambda(-0.5) = n + 0.5\lambda$. Since $0.5\lambda > 0$, $n_g > n$.