The Refraction Formula, expressed as n₁ sin θ₁ = n₂ sin θ₂ (Snell’s Law), describes how light bends when it travels from one transparent medium into another of different optical density. It is a core concept in NCERT Class 10 Science (Chapter 10) and Class 12 Physics (Chapter 9), and it appears regularly in CBSE board exams, JEE Main, JEE Advanced, and NEET. This article covers the complete refraction formula, its derivation, a full formula sheet, three progressive solved examples, CBSE exam tips for 2025-26, common mistakes, and JEE/NEET applications.
Key Refraction Formulas at a Glance
Quick reference for the most important refraction formulas used in CBSE and competitive exams.
- Snell’s Law: \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \)
- Refractive index (speed): \( n = \dfrac{c}{v} \)
- Refractive index (ratio of sines): \( n_{21} = \dfrac{\sin\theta_1}{\sin\theta_2} \)
- Critical angle: \( \sin\theta_c = \dfrac{n_2}{n_1} \) (for \( n_1 > n_2 \))
- Lens maker’s equation: \( \dfrac{1}{f} = (n-1)\left(\dfrac{1}{R_1} – \dfrac{1}{R_2}\right) \)
- Refraction at a spherical surface: \( \dfrac{n_2}{v} – \dfrac{n_1}{u} = \dfrac{n_2 – n_1}{R} \)
- Apparent depth: \( d’ = \dfrac{d}{n} \)
What is the Refraction Formula?
The Refraction Formula gives a precise mathematical relationship between the angles of incidence and refraction when light crosses the boundary between two media. When light travels through a uniform medium, it moves in a straight line. However, when it enters a second medium with a different optical density, its speed changes. This change in speed causes the ray to bend — a phenomenon called refraction.
The bending is governed by Snell’s Law, named after Dutch mathematician Willebrord Snellius. In NCERT Class 10 Science, Chapter 10 (“Light — Reflection and Refraction”), students first encounter this concept. It is revisited in depth in NCERT Class 12 Physics, Chapter 9 (“Ray Optics and Optical Instruments”). The refractive index of a medium determines how much light slows down and bends inside it. A higher refractive index means a denser optical medium and greater bending of light.
Two key laws govern refraction: (1) the incident ray, refracted ray, and normal all lie in the same plane; (2) the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for a given pair of media. This second law is the Refraction Formula.
Refraction Formula — Expression and Variables
The standard form of the Refraction Formula (Snell’s Law) is:
\[ n_1 \sin\theta_1 = n_2 \sin\theta_2 \]
An equivalent and commonly used form expresses the relative refractive index of medium 2 with respect to medium 1:
\[ n_{21} = \frac{n_2}{n_1} = \frac{\sin\theta_1}{\sin\theta_2} \]
The refractive index of a medium in terms of the speed of light is:
\[ n = \frac{c}{v} \]
| Symbol | Quantity | SI Unit / Value |
|---|---|---|
| \( n_1 \) | Refractive index of medium 1 (incident medium) | Dimensionless |
| \( n_2 \) | Refractive index of medium 2 (refracted medium) | Dimensionless |
| \( \theta_1 \) | Angle of incidence (measured from normal) | Degrees or Radians |
| \( \theta_2 \) | Angle of refraction (measured from normal) | Degrees or Radians |
| \( c \) | Speed of light in vacuum | 3 × 10⁸ m/s |
| \( v \) | Speed of light in the medium | m/s |
| \( n_{21} \) | Refractive index of medium 2 w.r.t. medium 1 | Dimensionless |
Derivation of the Refraction Formula
Consider a plane wavefront travelling from medium 1 (refractive index \( n_1 \)) to medium 2 (refractive index \( n_2 \)). Using Huygens’ wave theory, each point on the wavefront acts as a secondary source.
Step 1: Let the wavefront strike the interface at point A. The part of the wavefront at A enters medium 2 immediately, while the part at B is still in medium 1.
Step 2: In time \( t \), the wavefront at B travels a distance \( v_1 t = BC \) in medium 1, while the secondary wavelet from A travels \( v_2 t = AD \) in medium 2.
Step 3: From the geometry of the triangles ABC and ACD:
\[ \sin\theta_1 = \frac{BC}{AC} = \frac{v_1 t}{AC}, \quad \sin\theta_2 = \frac{AD}{AC} = \frac{v_2 t}{AC} \]
Step 4: Dividing: \( \dfrac{\sin\theta_1}{\sin\theta_2} = \dfrac{v_1}{v_2} \). Since \( n = c/v \), we get \( \dfrac{v_1}{v_2} = \dfrac{n_2}{n_1} \), which gives:
\[ n_1 \sin\theta_1 = n_2 \sin\theta_2 \]
This is Snell’s Law — the Refraction Formula.
Complete Optics Formula Sheet
| Formula Name | Expression | Variables | SI Units | NCERT Chapter |
|---|---|---|---|---|
| Snell’s Law (Refraction Formula) | \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \) | n = refractive index, θ = angle with normal | Dimensionless | Class 12, Ch 9 / Class 10, Ch 10 |
| Refractive Index (speed) | \( n = c / v \) | c = 3×10⁸ m/s, v = speed in medium | Dimensionless | Class 12, Ch 9 |
| Critical Angle | \( \sin\theta_c = n_2 / n_1 \) | θc = critical angle, n₁ > n₂ | Degrees | Class 12, Ch 9 |
| Apparent Depth | \( d’ = d / n \) | d = real depth, n = refractive index | m | Class 10, Ch 10 |
| Refraction at Spherical Surface | \( \dfrac{n_2}{v} – \dfrac{n_1}{u} = \dfrac{n_2 – n_1}{R} \) | u = object distance, v = image distance, R = radius | m | Class 12, Ch 9 |
| Lens Maker’s Equation | \( \dfrac{1}{f} = (n-1)\left(\dfrac{1}{R_1} – \dfrac{1}{R_2}\right) \) | f = focal length, R₁, R₂ = radii of curvature | m | Class 12, Ch 9 |
| Thin Lens Formula | \( \dfrac{1}{v} – \dfrac{1}{u} = \dfrac{1}{f} \) | u = object distance, v = image distance, f = focal length | m | Class 12, Ch 9 |
| Mirror Formula | \( \dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f} \) | u = object distance, v = image distance, f = focal length | m | Class 12, Ch 9 |
| Power of a Lens | \( P = 1/f \) | f = focal length in metres | Dioptre (D) | Class 12, Ch 9 |
| Prism Formula (minimum deviation) | \( n = \dfrac{\sin\left(\frac{A+\delta_m}{2}\right)}{\sin(A/2)} \) | A = prism angle, δm = minimum deviation | Dimensionless | Class 12, Ch 9 |
Refraction Formula — Solved Examples
Example 1 (Class 10 Level)
Problem: A ray of light travels from air into glass. The angle of incidence is 30°. The refractive index of glass is 1.5. Find the angle of refraction.
Given: \( n_1 = 1 \) (air), \( n_2 = 1.5 \) (glass), \( \theta_1 = 30° \)
Step 1: Write the Refraction Formula (Snell’s Law):
\( n_1 \sin\theta_1 = n_2 \sin\theta_2 \)
Step 2: Substitute the known values:
\( 1 \times \sin 30° = 1.5 \times \sin\theta_2 \)
Step 3: Recall that \( \sin 30° = 0.5 \):
\( 0.5 = 1.5 \times \sin\theta_2 \)
Step 4: Solve for \( \sin\theta_2 \):
\( \sin\theta_2 = \dfrac{0.5}{1.5} = 0.333 \)
Step 5: Find the angle:
\( \theta_2 = \sin^{-1}(0.333) \approx 19.47° \)
Answer
The angle of refraction is approximately 19.5°. The ray bends towards the normal as it enters the denser medium (glass).
Example 2 (Class 12 Level)
Problem: Light travels from water (refractive index 4/3) into a glass slab (refractive index 3/2). The angle of incidence at the water-glass interface is 45°. Find the angle of refraction inside the glass.
Given: \( n_1 = 4/3 \approx 1.333 \) (water), \( n_2 = 3/2 = 1.5 \) (glass), \( \theta_1 = 45° \)
Step 1: Apply the Refraction Formula:
\( n_1 \sin\theta_1 = n_2 \sin\theta_2 \)
Step 2: Substitute values. Note \( \sin 45° = 1/\sqrt{2} \approx 0.7071 \):
\( \dfrac{4}{3} \times \dfrac{1}{\sqrt{2}} = \dfrac{3}{2} \times \sin\theta_2 \)
Step 3: Simplify the left side:
\( \dfrac{4}{3\sqrt{2}} = \dfrac{3}{2} \times \sin\theta_2 \)
Step 4: Solve for \( \sin\theta_2 \):
\( \sin\theta_2 = \dfrac{4}{3\sqrt{2}} \times \dfrac{2}{3} = \dfrac{8}{9\sqrt{2}} = \dfrac{8}{12.728} \approx 0.6285 \)
Step 5: Find the angle:
\( \theta_2 = \sin^{-1}(0.6285) \approx 38.94° \)
Observation: The ray bends towards the normal because it moves from a less dense to a more dense optical medium.
Answer
The angle of refraction inside the glass is approximately 38.9°.
Example 3 (JEE/NEET Level)
Problem: A ray of light strikes the flat surface of a glass slab (refractive index \( n = \sqrt{3} \)) from inside at an angle of incidence of 40°. Determine whether total internal reflection occurs. Also, find the critical angle for this glass-air interface.
Given: \( n_1 = \sqrt{3} \approx 1.732 \) (glass), \( n_2 = 1 \) (air), \( \theta_i = 40° \)
Step 1: Calculate the critical angle using:
\( \sin\theta_c = \dfrac{n_2}{n_1} = \dfrac{1}{\sqrt{3}} \approx 0.5774 \)
Step 2: Find \( \theta_c \):
\( \theta_c = \sin^{-1}(0.5774) \approx 35.26° \)
Step 3: Compare the angle of incidence with the critical angle:
\( \theta_i = 40° > \theta_c = 35.26° \)
Step 4: Since the angle of incidence exceeds the critical angle, total internal reflection occurs. The ray does not refract into air; it reflects entirely back into the glass.
Step 5: Verify using the Refraction Formula. Attempt \( \sin\theta_2 = \dfrac{n_1}{n_2} \sin\theta_i = \sqrt{3} \times \sin 40° \approx 1.732 \times 0.6428 = 1.113 \). Since \( \sin\theta_2 > 1 \), no refracted ray exists — confirming total internal reflection.
Answer
The critical angle is approximately 35.3°. Since the angle of incidence (40°) exceeds the critical angle, total internal reflection occurs and no refracted ray exits the glass.
CBSE Exam Tips 2025-26
- Always measure angles from the normal, not the surface. A very common error is measuring θ from the interface itself. The normal is perpendicular to the surface at the point of incidence.
- Memorise standard refractive indices. CBSE frequently provides these values in numerical problems: air ≈ 1.0003 (taken as 1), water = 4/3, glass = 3/2, diamond ≈ 2.42. We recommend writing these on your formula sheet.
- State both laws of refraction in theory questions. In 2-mark or 3-mark questions, examiners expect you to state both laws. Write them clearly before applying the formula.
- For apparent depth problems, use \( d’ = d/n \) only when the observer is in air and the object is in a denser medium directly below. If the geometry changes, revert to the full Snell’s Law treatment.
- Draw a clear ray diagram. In CBSE 2025-26 exams, diagrams carry separate marks. Label the incident ray, refracted ray, normal, and both angles clearly.
- Practice reversibility. The path of light is reversible. If light goes from medium 1 to medium 2 with angle θ₁ → θ₂, then light going from medium 2 to medium 1 follows θ₂ → θ₁. This is useful in multi-step problems.
Common Mistakes to Avoid
- Measuring angles from the surface instead of the normal. The correct approach is always to measure the angle of incidence and angle of refraction from the normal drawn at the point of incidence. Measuring from the surface gives the complement of the correct angle and leads to a completely wrong answer.
- Confusing refractive index direction. Students often write \( n_{21} = n_1/n_2 \) instead of \( n_{21} = n_2/n_1 \). Remember: \( n_{21} \) is the refractive index of medium 2 with respect to medium 1, so medium 2 goes in the numerator.
- Applying total internal reflection in the wrong direction. Total internal reflection only occurs when light travels from a denser medium to a rarer medium (e.g., glass to air), not the other way around. Light going from air to glass cannot undergo total internal reflection.
- Ignoring units for angles. Snell’s Law uses sine values of angles. Always ensure your calculator is set to degrees (not radians) when working with angles given in degrees. This is a surprisingly frequent error in exams.
- Forgetting that refractive index is always ≥ 1 for a medium relative to vacuum. Some students write n < 1 for a medium, which is physically incorrect under normal conditions. The refractive index of any medium with respect to vacuum is always greater than or equal to 1.
JEE/NEET Application of Refraction Formula
In our experience, JEE aspirants encounter the Refraction Formula in nearly every optics problem. It rarely appears in isolation. Understanding how to apply it in multi-interface and multi-step situations is essential for scoring well.
Application Pattern 1: Multi-layer Refraction
A common JEE problem involves light passing through several parallel slabs. The key insight is that if the first and last media are the same (e.g., both air), then the emergent ray is parallel to the incident ray. Mathematically, applying Snell’s Law at each interface and multiplying through gives:
\[ n_1 \sin\theta_1 = n_2 \sin\theta_2 = n_3 \sin\theta_3 = \cdots \]
This means the product \( n \sin\theta \) is conserved across all parallel interfaces. This shortcut saves enormous time in JEE problems.
Application Pattern 2: Prism and Minimum Deviation
The prism formula is a direct consequence of the Refraction Formula applied twice — once at each refracting surface. At minimum deviation, the ray travels symmetrically through the prism. The refractive index is then:
\[ n = \frac{\sin\left(\dfrac{A + \delta_m}{2}\right)}{\sin\left(\dfrac{A}{2}\right)} \]
NEET frequently tests this formula with numerical values of A and δm. JEE Advanced tests the derivation itself.
Application Pattern 3: Optical Fibre and Total Internal Reflection
Optical fibres use total internal reflection — a direct consequence of the Refraction Formula when the angle of incidence exceeds the critical angle. JEE problems often ask for the acceptance angle or numerical aperture of a fibre. The critical angle \( \theta_c \) satisfies \( \sin\theta_c = n_2/n_1 \), and the numerical aperture is \( \text{NA} = \sqrt{n_1^2 – n_2^2} \). Our experts suggest practising at least 10 such problems before the exam to build confidence.
FAQs on Refraction Formula
For a deeper understanding of related optics concepts, explore our detailed articles on the Physics Formulas hub, which covers the complete NCERT Class 12 Physics syllabus. You may also find our articles on the Critical Velocity Formula and the Induced Voltage Formula useful for building a strong foundation in physics. For the official NCERT curriculum reference, visit ncert.nic.in.