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Ray Optics Fundamentals

Snell's Law

Explore how light waves change direction when crossing boundaries between different media. Practice incident adjustments, analyze angular sines, and observe critical refraction and total internal reflection inside our virtual laboratory.

Snell's Law Interactive Laboratory ⓘ

Animate the incident laser to verify that n₁ sin i = n₂ sin r is constant across boundaries.

Simulating...

Live Telemetry

Snell's Law: n₁ sin i = n₂ sin r
Preset Media Pair
Air → Glass
Indices (n₁ / n₂)
1.00 / 1.50
Incident Angle (i)
45.0°
Refracted Angle (r)
28.1°
Sines (sin i / sin r)
0.707 / 0.471
Bending Direction
Toward Normal

What is Snell's Law?

**Snell's Law** (also called the Snell-Descartes law or the second law of refraction) is a fundamental physics equation that describes the relationship between the angles of incidence and refraction when a light wave crosses the boundary between two transparent media with different refractive indices.

The Mathematical Formula

Snell's Law states that for a given pair of media, the ratio of the sines of the angle of incidence ($i$) and angle of refraction ($r$) is equal to the ratio of the refractive indices of the two media ($n_1$ and $n_2$):

n_1 \sin i = n_2 \sin r

Where:

  • **$n_1$**: Refractive index of the medium from which light originates (incident medium).
  • **$n_2$**: Refractive index of the medium into which light enters (refracted medium).
  • **$i$**: Angle of incidence (measured between the incident ray and the normal line).
  • **$r$**: Angle of refraction (measured between the refracted ray and the normal line).

Critical Angle and Total Internal Reflection (TIR)

When light travels from an optically denser medium to a rarer medium ($n_1 \gt n_2$), it bends away from the normal, meaning $r \gt i$. As the angle of incidence increases, there comes a point where the angle of refraction reaches exactly $90^\circ$ (propagating parallel to the interface). The incident angle that causes this is called the **critical angle** ($\theta_c$):

\sin \theta_c = \frac{n_2}{n_1}

If the angle of incidence is increased further ($\theta_i \gt \theta_c$), refraction is no longer possible, and all light reflects entirely back inside the incident medium. This phenomenon is known as **Total Internal Reflection** (TIR).

Wave Speed and Wavelength Relations

Because the refractive index of a medium is defined as $n = c/v$, where $v$ is the phase speed of light inside the medium, Snell's Law can also be written in terms of speed and wavelength:

\frac{\sin i}{\sin r} = \frac{n_2}{n_1} = \frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2}

This illustrates that when light enters a denser medium (higher $n$), it slows down ($v_2 \lt v_1$) and its wavelength compresses ($\lambda_2 \lt \lambda_1$), while its frequency ($f$) remains constant.

Solved Examples

Example 1

A ray of light traveling in air strikes the flat surface of a rectangular glass block at an angle of 30.0° relative to the normal line. If the refractive index of the glass is 1.50, calculate the angle of refraction inside the block.

View Step-by-Step Solution
  1. Identify the given values: Refractive index of air n₁ ≈ 1.00, refractive index of glass n₂ = 1.50, and incident angle i = 30.0°.
  2. Recall Snell's Law: n₁ sin i = n₂ sin r.
  3. Rearrange the formula to solve for sin r: sin r = (n₁ sin i) / n₂.
  4. Substitute values: sin r = (1.00 × sin 30.0°) / 1.50.
  5. Calculate sine of 30°: sin 30° = 0.50. Thus, sin r = (1.00 × 0.50) / 1.50 = 0.3333.
  6. Compute the inverse sine (arcsin) to find the angle r: r = arcsin(0.3333) ≈ 19.5°.
  7. The light bends toward the normal line, narrowing its angle from 30° down to 19.5°.

Final Answer: Angle of Refraction in Glass ≈ 19.5°

Example 2

Find the critical angle for a boundary between water (n₁ = 1.33) and air (n₂ = 1.00), explaining what happens if the incident angle inside water exceeds this value.

View Step-by-Step Solution
  1. Identify the given values: refractive index of water n₁ = 1.33, refractive index of air n₂ = 1.00.
  2. Define the critical angle condition: The angle of refraction r reaches exactly 90.0° (sin 90° = 1.00).
  3. Recall Snell's Law: n₁ sin θ_c = n₂ sin 90°.
  4. Simplify to solve for sin θ_c: sin θ_c = n₂ / n₁.
  5. Substitute values: sin θ_c = 1.00 / 1.33 ≈ 0.750.
  6. Compute the inverse sine to find the critical angle: θ_c = arcsin(0.750) ≈ 48.8°.
  7. If the incident angle i exceeds 48.8° inside water, refraction becomes mathematically impossible (sin r > 1.0) and all light reflects back inside water. This is Total Internal Reflection (TIR).

Final Answer: Critical Angle for Water-Air Boundary ≈ 48.8°

Example 3

A light ray inside a solid plastic block (n = 1.42) strikes the boundary with water (n = 1.33) at an angle of incidence of 65.0°. Determine whether the light refracts into the water, and calculate the angle of refraction if it does.

View Step-by-Step Solution
  1. Identify the given values: n₁ = 1.42 (plastic), n₂ = 1.33 (water), and incident angle i = 65.0°.
  2. Calculate the critical angle for this specific boundary: sin θ_c = n₂ / n₁ = 1.33 / 1.42 ≈ 0.9366.
  3. Find θ_c: θ_c = arcsin(0.9366) ≈ 69.5°.
  4. Compare the incident angle to the critical angle: Since the incident angle (65.0°) is less than the critical angle (69.5°), the light WILL refract into the water.
  5. Set up Snell's Law to find the angle of refraction: n₁ sin i = n₂ sin r.
  6. Rearrange: sin r = (n₁ sin i) / n₂ = (1.42 × sin 65.0°) / 1.33.
  7. Calculate: sin 65.0° ≈ 0.9063, so sin r = (1.42 × 0.9063) / 1.33 ≈ 0.9676.
  8. Find r: r = arcsin(0.9676) ≈ 75.4°.
  9. Because light enters a less optically dense medium (n₂ < n₁), it bends away from the normal, increasing its angle from 65.0° to 75.4°.

Final Answer: The ray refracts into water; Angle of Refraction ≈ 75.4°

Self-Check Questions

Question 1

State the physical reason why light changes direction when entering a new medium.

Show Answer & Explanation

Refraction occurs because light propagates at different speeds in different media. When a light wave strikes a boundary at an angle, one side of the wavefront slows down or speeds up before the other side, causing the entire wavefront to tilt or bend. This speed change is quantified by the material's refractive index (n = c / v).

Question 2

What is the mathematical relation between the speed of light in two media and their angles of refraction?

Show Answer & Explanation

Combining the definitions of refractive index (n = c / v) and Snell's Law (n₁ sin i = n₂ sin r), we get: (c / v₁) sin i = (c / v₂) sin r. Canceling the speed of light in vacuum (c) gives the speed-angle relation: sin i / sin r = v₁ / v₂. This shows that the sine of the angles is directly proportional to light speed in the respective media.

Question 3

Explain what is meant by the 'principle of reversibility of light' in the context of Snell's Law.

Show Answer & Explanation

The principle of reversibility states that if the path of a ray of light is reversed, it will retrace its path exactly. Mathematically, Snell's Law n₁ sin i = n₂ sin r is symmetric. If light goes from medium 1 to 2 at angle i, refracting at r, it will also travel from medium 2 to 1 at incident angle r, refracting at angle i.