The Refractive Index Formula gives the ratio of the speed of light in a vacuum to its speed in a given medium, expressed as n = c/v. It is a dimensionless quantity covered in NCERT Class 10 (Chapter 10) and Class 12 (Chapter 9) Physics. For JEE Main and NEET aspirants, this formula is essential for solving problems on refraction, Snell's Law, total internal reflection, and optical instruments. This article covers the complete formula, derivation, a full formula sheet, three solved examples at progressive difficulty levels, CBSE exam tips, common mistakes, and JEE/NEET applications.
Key Refractive Index Formulas at a Glance
Quick reference for the most important refractive index formulas.
- Absolute refractive index: \( n = \dfrac{c}{v} \)
- Snell's Law: \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \)
- Relative refractive index: \( _1n_2 = \dfrac{n_2}{n_1} = \dfrac{v_1}{v_2} \)
- Critical angle relation: \( \sin C = \dfrac{1}{n} \)
- Refractive index via wavelength: \( n = \dfrac{\lambda_0}{\lambda_m} \)
- Lens Maker's equation link: \( \frac{1}{f} = (n-1)\left(\frac{1}{R_1} – \frac{1}{R_2}\right) \)
- Cauchy's relation (dispersion): \( n = A + \dfrac{B}{\lambda^2} \)
What is the Refractive Index Formula?
The Refractive Index Formula defines how much a medium slows down light compared to its speed in a vacuum. When light travels from one medium to another, it changes speed. This change in speed causes the ray to bend — a phenomenon called refraction. The refractive index quantifies this behaviour.
In NCERT Class 10 Physics, Chapter 10 (Light — Reflection and Refraction), students encounter the basic definition. Class 12 Physics, Chapter 9 (Ray Optics and Optical Instruments), extends this to Snell's Law, critical angle, and total internal reflection. The refractive index is a dimensionless, unitless quantity. It is always greater than or equal to 1 for all known natural media, since light travels fastest in a vacuum.
For visible light, most optical media have refractive indices between 1.0 (vacuum) and 2.5 (diamond). Infrared radiation can encounter significantly higher values in certain materials. In optical microscopy, the refractive index determines the numerical aperture of a lens system. Understanding this formula is therefore fundamental to optics, both at the school level and in competitive examinations.
Refractive Index Formula — Expression and Variables
The absolute refractive index of a medium is defined as:
\[ n = \frac{c}{v} \]
where c is the speed of light in vacuum and v is the speed of light in the medium.
Snell's Law connects the refractive indices of two media at an interface:
\[ n_1 \sin\theta_1 = n_2 \sin\theta_2 \]
The relative refractive index of medium 2 with respect to medium 1 is:
\[ _1n_2 = \frac{n_2}{n_1} = \frac{v_1}{v_2} = \frac{\sin\theta_1}{\sin\theta_2} \]
| Symbol | Quantity | SI Unit / Nature |
|---|---|---|
| \( n \) | Absolute refractive index | Dimensionless (no unit) |
| \( c \) | Speed of light in vacuum | m/s (\( 3 \times 10^8 \) m/s) |
| \( v \) | Speed of light in the medium | m/s |
| \( n_1 \) | Refractive index of medium 1 | Dimensionless |
| \( n_2 \) | Refractive index of medium 2 | Dimensionless |
| \( \theta_1 \) | Angle of incidence (in medium 1) | Degrees or Radians |
| \( \theta_2 \) | Angle of refraction (in medium 2) | Degrees or Radians |
| \( \lambda_0 \) | Wavelength of light in vacuum | m (or nm) |
| \( \lambda_m \) | Wavelength of light in the medium | m (or nm) |
| \( C \) | Critical angle | Degrees |
Derivation of the Refractive Index Formula
The refractive index arises from the electromagnetic theory of light. Light is an electromagnetic wave. Its speed in any medium depends on the permittivity (\( \varepsilon \)) and permeability (\( \mu \)) of that medium.
Step 1: Speed of light in vacuum: \( c = \dfrac{1}{\sqrt{\varepsilon_0 \mu_0}} \)
Step 2: Speed of light in medium: \( v = \dfrac{1}{\sqrt{\varepsilon \mu}} \)
Step 3: Taking the ratio: \( n = \dfrac{c}{v} = \sqrt{\dfrac{\varepsilon \mu}{\varepsilon_0 \mu_0}} = \sqrt{\varepsilon_r \mu_r} \)
Step 4: For most optical materials, \( \mu_r \approx 1 \), so \( n \approx \sqrt{\varepsilon_r} \). This is the Maxwell relation.
Snell's Law is derived from Huygens' Principle by applying the condition that wavefronts must be continuous at the interface, leading to \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \).
Complete Optics Formula Sheet
| Formula Name | Expression | Variables | SI Units | NCERT Chapter |
|---|---|---|---|---|
| Absolute Refractive Index | \( n = c/v \) | c = speed in vacuum, v = speed in medium | Dimensionless | Class 10, Ch 10; Class 12, Ch 9 |
| Snell's Law | \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \) | n = refractive index, \( \theta \) = angle with normal | Dimensionless | Class 12, Ch 9 |
| Relative Refractive Index | \( _1n_2 = n_2/n_1 = v_1/v_2 \) | v = speed in respective medium | Dimensionless | Class 12, Ch 9 |
| Critical Angle Formula | \( \sin C = 1/n \) | C = critical angle, n = refractive index (denser medium) | Degrees | Class 12, Ch 9 |
| Refractive Index via Wavelength | \( n = \lambda_0 / \lambda_m \) | \( \lambda_0 \) = wavelength in vacuum, \( \lambda_m \) = in medium | Dimensionless | Class 12, Ch 9 |
| Lens Maker's Equation | \( \frac{1}{f} = (n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right) \) | f = focal length, R = radii of curvature, n = refractive index | m⁻¹ (for f) | Class 12, Ch 9 |
| Cauchy's Relation (Dispersion) | \( n = A + B/\lambda^2 \) | A, B = Cauchy constants, \( \lambda \) = wavelength | Dimensionless | Class 12, Ch 9 |
| Brewster's Angle | \( \tan\theta_B = n \) | \( \theta_B \) = Brewster's angle, n = refractive index | Degrees | Class 12, Ch 10 |
| Apparent Depth Formula | \( n = \text{Real Depth} / \text{Apparent Depth} \) | n = refractive index of denser medium | Dimensionless | Class 10, Ch 10 |
| Refraction at Spherical Surface | \( \frac{n_2}{v} – \frac{n_1}{u} = \frac{n_2 – n_1}{R} \) | u = object distance, v = image distance, R = radius | m (for distances) | Class 12, Ch 9 |
Refractive Index Formula — Solved Examples
Example 1 (Class 9-10 Level)
Problem: The speed of light in glass is \( 2 \times 10^8 \) m/s. Find the refractive index of glass. (Speed of light in vacuum = \( 3 \times 10^8 \) m/s.)
Given: \( c = 3 \times 10^8 \) m/s, \( v = 2 \times 10^8 \) m/s
Step 1: Write the Refractive Index Formula: \( n = \dfrac{c}{v} \)
Step 2: Substitute the values: \( n = \dfrac{3 \times 10^8}{2 \times 10^8} \)
Step 3: Simplify: \( n = \dfrac{3}{2} = 1.5 \)
Answer
The refractive index of glass = 1.5 (dimensionless).
Example 2 (Class 11-12 Level)
Problem: A ray of light travels from air (\( n_1 = 1.0 \)) into water (\( n_2 = 1.33 \)). The angle of incidence is 45°. Find the angle of refraction.
Given: \( n_1 = 1.0 \), \( n_2 = 1.33 \), \( \theta_1 = 45° \)
Step 1: Apply Snell's Law: \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \)
Step 2: Substitute values: \( 1.0 \times \sin 45° = 1.33 \times \sin\theta_2 \)
Step 3: Calculate \( \sin 45° = 0.7071 \): \( 1.0 \times 0.7071 = 1.33 \times \sin\theta_2 \)
Step 4: Solve for \( \sin\theta_2 \): \( \sin\theta_2 = \dfrac{0.7071}{1.33} = 0.5317 \)
Step 5: Find \( \theta_2 \): \( \theta_2 = \sin^{-1}(0.5317) \approx 32.1° \)
Answer
The angle of refraction in water \( \approx \) 32.1°. The ray bends towards the normal as it enters a denser medium, which confirms the result.
Example 3 (JEE/NEET Level)
Problem: A glass slab of refractive index 1.5 and thickness 9 cm is placed in air. A point object is placed 12 cm in front of the slab. Find the apparent shift in the position of the object as seen from the other side of the slab.
Given: \( n = 1.5 \), real thickness \( t = 9 \) cm, object distance = 12 cm
Step 1: Recall the apparent depth formula. For a slab of thickness \( t \) and refractive index \( n \), the apparent thickness is \( t_{app} = t/n \).
Step 2: Calculate apparent thickness: \( t_{app} = \dfrac{9}{1.5} = 6 \) cm
Step 3: Calculate the apparent shift (normal shift): \( \Delta = t – t_{app} = 9 – 6 = 3 \) cm
Step 4: The object appears to shift towards the observer by 3 cm. The apparent position of the object from the near face of the slab is \( 12 – 3 = 9 \) cm (as seen from the far side, the object appears 3 cm closer).
Step 5: Verify using the formula: Normal shift \( = t\left(1 – \dfrac{1}{n}\right) = 9\left(1 – \dfrac{1}{1.5}\right) = 9 \times \dfrac{1}{3} = 3 \) cm ✓
Answer
The apparent shift in the position of the object = 3 cm (object appears 3 cm closer to the observer).
CBSE Exam Tips 2025-26
- Always measure angles from the normal: A very common error is measuring angles from the surface instead of the normal. In CBSE 2025-26 board exams, this mistake costs full marks on Snell's Law problems.
- State the formula before substituting: CBSE marking schemes award one mark for writing the correct formula. We recommend always writing \( n = c/v \) or \( n_1\sin\theta_1 = n_2\sin\theta_2 \) before plugging in numbers.
- Remember that refractive index has no unit: Many students write “n = 1.5 m/s” by mistake. The refractive index is dimensionless. Mentioning the unit will cost you marks.
- Learn standard values by heart: The CBSE board frequently uses standard values — glass (1.5), water (1.33), diamond (2.42), ice (1.31). Memorise these for quick substitution.
- Link critical angle to refractive index: Questions on total internal reflection always involve \( \sin C = 1/n \). Our experts suggest practising at least 5 problems combining Snell's Law with the critical angle formula before the exam.
- Check for the denser medium: Total internal reflection occurs only when light travels from a denser medium to a rarer medium. Always identify which medium is denser before applying the critical angle formula.
Common Mistakes to Avoid
| Common Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| Using angle from surface instead of normal | Snell's Law requires angles measured from the normal at the point of incidence | Always subtract the given angle from 90° if it is measured from the surface |
| Writing a unit for refractive index | Refractive index is a ratio of two speeds; units cancel out | State “n = 1.5 (dimensionless)” or simply “n = 1.5” |
| Confusing absolute and relative refractive index | Absolute n is relative to vacuum; relative \( _1n_2 \) is medium 2 relative to medium 1 | Check whether the problem gives speed in vacuum or in another medium |
| Applying total internal reflection for rarer-to-denser travel | TIR is only possible when light moves from denser to rarer medium | Identify the optical density of each medium before applying the concept |
| Forgetting that frequency does not change on refraction | Only speed and wavelength change; frequency remains constant | Use \( n = \lambda_0/\lambda_m \) and remember \( v = f\lambda \) to avoid confusion |
JEE/NEET Application of Refractive Index Formula
In our experience, JEE aspirants encounter the Refractive Index Formula in almost every optics problem. The formula connects directly to Snell's Law, total internal reflection, prism deviation, and optical fibre principles — all high-weightage topics.
Pattern 1: Prism and Angle of Deviation
JEE Main frequently tests the minimum deviation condition for a prism. The refractive index of a prism is given by:
\[ n = \frac{\sin\left(\dfrac{A + D_m}{2}\right)}{\sin\left(\dfrac{A}{2}\right)} \]
where \( A \) is the prism angle and \( D_m \) is the minimum angle of deviation. Students must apply Snell's Law at both faces of the prism and use the symmetry condition at minimum deviation.
Pattern 2: Optical Fibre and Total Internal Reflection
NEET and JEE both test the concept of optical fibres. The acceptance angle \( \theta_a \) and numerical aperture (NA) are derived directly from the refractive index:
\[ NA = \sin\theta_a = \sqrt{n_1^2 – n_2^2} \]
where \( n_1 \) is the refractive index of the core and \( n_2 \) is that of the cladding. In our experience, JEE aspirants who master the critical angle formula \( \sin C = n_2/n_1 \) can solve these problems in under a minute.
Pattern 3: Apparent Depth and Normal Shift
NEET Biology-linked Physics questions often involve a fish seeing an object above water or a person looking at a coin in a glass. The apparent depth formula \( d_{app} = d_{real}/n \) and the normal shift formula \( \Delta = t(1 – 1/n) \) are both direct applications of the Refractive Index Formula. These appear in 1-2 questions per NEET paper on average.
We recommend practising problems from NCERT Exemplar Class 12 Chapter 9 and previous 10 years of JEE Main papers to master all three patterns.
FAQs on Refractive Index Formula
Explore More Physics Formulas
Now that you have mastered the Refractive Index Formula, strengthen your optics and physics preparation with these related resources on ncertbooks.net:
- Visit our complete Physics Formulas hub for a full list of NCERT-aligned formula articles.
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- Explore the Induced Voltage Formula to connect electromagnetic wave theory with the origin of light itself.
- Review the Superposition Formula to understand how light waves combine — directly relevant to interference and diffraction optics.
For the official NCERT syllabus and chapter references, visit the NCERT official website.