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Snell's law of refraction

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Snell's law describes how much light bends when it crosses a boundary between two materials: n₁ sin θ₁ = n₂ sin θ₂. Here n₁ and n₂ are the refractive indices of the two materials, and θ₁ and θ₂ are the angles the ray makes with the normal on each side. When light goes from air into a material with refractive index n, it simplifies to n = sin i ÷ sin r. Light entering glass (n = 1.52) at 40° to the normal bends to 25.0°.

The Snell's law formula

n_1\sin\theta_1 = n_2\sin\theta_2Snell's law of refraction for a boundary
SymbolMeaning
n₁Refractive index of the first material, where the light starts
θ₁Angle of incidence, between the incoming ray and the normal
n₂Refractive index of the second material
θ₂Angle of refraction, between the refracted ray and the normal

The normal is an imaginary line at right angles to the boundary, where the ray meets it. AQA's A-level data booklet gives Snell's law in this form, and AQA's specification expects students to recall that the refractive index of air is approximately 1.

From air into a material

With n₁ = 1 for air, Snell's law becomes sin i = n sin r, which rearranges to:

n = \frac{\sin i}{\sin r}

where i is the angle in air and r the angle in the material. This is the form often used at GCSE and in the refraction practical with a glass block.

40°25°airglass
Into a higher refractive index, the ray bends towards the normal

Refractive index

The refractive index of a material tells you how much it slows light down:

n = \frac{c}{c_s}

where c is the speed of light in a vacuum, 3.00 × 10⁸ m/s, and cs is its speed in the substance. AQA's data booklet gives this as refractive index of a substance s. Light is slower in water and glass than in air, so their refractive indices are bigger than 1.

MaterialRefractive index
Ice1.31
Water at 20 °C1.33
Fused quartz1.46
Typical crown glass1.52
Diamond2.417

These values are from the HyperPhysics table at Georgia State University.

Bending towards or away from the normal

  • Into a higher refractive index (air into glass), the light slows down and bends towards the normal: θ₂ is smaller than θ₁.
  • Into a lower refractive index (glass into air), the light speeds up and bends away from the normal: θ₂ is bigger than θ₁.
  • Along the normal (θ₁ = 0), the light does not bend at all, though it still changes speed.

Snell's law worked examples

Example 1: air into glass

Light in air hits a glass block (n = 1.52) at 40° to the normal. Find the angle of refraction.

\sin\theta_2 = \frac{1.00 \times \sin 40°}{1.52} = 0.423, \quad \theta_2 = 25.0°

Example 2: finding the refractive index

In a practical, a ray enters a plastic block at 50° and refracts to 30°. Find the refractive index.

n = \frac{\sin 50°}{\sin 30°} = \frac{0.766}{0.500} = 1.53

Example 3: water into glass

Light travelling in water (n = 1.33) meets glass (n = 1.52) at 35° to the normal. Find the angle of refraction.

\sin\theta_2 = \frac{1.33 \times \sin 35°}{1.52} = 0.502, \quad \theta_2 = 30.1°

It still bends towards the normal, though only by about 5°, since 1.33 and 1.52 differ much less than 1.00 and 1.52 do.

Example 4: glass into air

Light inside glass (n = 1.52) reaches the surface at 30° to the normal. Find the angle in air.

\sin\theta_2 = \frac{1.52 \times \sin 30°}{1.00} = 0.760, \quad \theta_2 = 49.5°

It bends away from the normal, from 30° to 49.5°.

Example 5: when Snell's law gives no answer

The same light reaches the surface at 45° instead. Then 1.52 × sin 45° = 1.07, but sin θ₂ cannot be more than 1. There is no refracted ray: the light is totally internally reflected. The angle where this starts is the critical angle, covered in our critical angle formula guide.

Example 6: the speed of light in water

c_s = \frac{c}{n} = \frac{3.00 \times 10^8}{1.33} = 2.26 \times 10^8\ \text{m/s}

Now try one in the course. Use Snell's law, then press Mark to check it there.

Calculate the angle of refraction for a random pair of materials and angle.

Press Mark to have your answer marked in the Light & Energy course.

The Refraction Calculator applies Snell's law in any direction, finds refractive indices and critical angles, and warns you when total internal reflection happens.

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Refraction Calculator

Use Snell's law to find angles, refractive index and the critical angle

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Measuring refractive index in a practical

Send a narrow beam from a ray box into the side of a glass block at a range of angles of incidence, mark where it enters and leaves, and join the marks to measure each angle of refraction. A graph of sin i against sin r should be a straight line that passes through (0, 0), with the refractive index as its gradient. Using a graph rather than one reading averages out errors in marking the rays.

Common mistakes

  • Measuring from the surface. Both angles are measured from the normal.
  • Putting the indices on the wrong sides. Each n goes with the angle in its own material: n₁ with θ₁, n₂ with θ₂.
  • Calculator in radians. Use degrees mode for sin and sin⁻¹.
  • Rounding sin θ too early. Keep three or four significant figures before taking sin⁻¹.
  • Forgetting total internal reflection. If n₁ sin θ₁ ÷ n₂ comes out bigger than 1, the light does not refract out.

Snell's law questions with answers

1. Light in air enters water (n = 1.33) at 60° to the normal. Find the angle of refraction.
sin θ₂ = sin 60° ÷ 1.33 = 0.651, so θ₂ = 40.6°
2. A ray enters a block at 45° and refracts to 28°. Find the refractive index.
n = sin 45° ÷ sin 28° = 1.51
3. Light in diamond (n = 2.417) meets air at 10° to the normal. Find the angle in air.
sin θ₂ = 2.417 × sin 10° = 0.420, so θ₂ = 24.8°
4. Light travels at 2.0 × 10⁸ m/s in a plastic. Find its refractive index.
n = 3.0 × 10⁸ ÷ 2.0 × 10⁸ = 1.5
5. Why does light not bend when it hits a boundary along the normal?
θ₁ = 0, so sin θ₁ = 0, and Snell's law gives θ₂ = 0. It changes speed but not direction.

Practise waves in a short timed paper

A-level Physics: Year 1 gives you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, so refraction turns up regularly but not in every paper. The Light and Energy course has an interactive Snell's law explorer.

Paper resets in

Common questions

What is Snell's law?
n₁ sin θ₁ = n₂ sin θ₂: it links the angles of incidence and refraction at a boundary to the refractive indices of the two materials.
What is the formula for refractive index?
n = c ÷ cs, the speed of light in a vacuum divided by its speed in the material. From air into a material, n = sin i ÷ sin r.
Does light bend towards or away from the normal?
Towards the normal when it enters a material with a higher refractive index, and away from the normal when it enters one with a lower refractive index.
What is the refractive index of air?
About 1. AQA's A-level specification expects students to recall this.
Why does light refract?
Its speed changes as it crosses the boundary. If it meets the boundary at an angle, one side of the wavefront changes speed first, so the direction changes.
Is Snell's law on the A-level data sheet?
Yes, in AQA's A-level data booklet, along with n = c ÷ cs and the critical angle formula. Check your own board's sheet.

Revision cards

Say the answer first, then flip the card to check it.

Where to go next

  • Solve it

    Refraction Calculator

    Use Snell's law to find angles, refractive index and the critical angle

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  • Learn it

    Light & Energy

    Waves, photons and the electromagnetic spectrum explained

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  • Test it

    A-level Physics: Year 1

    A new exam-style paper every day, with instant marking and a mark scheme

    Start today's paper
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