05.07 Snell's Law

The big idea Refraction changes a ray's direction because light travels at different speeds in different materials; Snell's law connects both angles to both refractive indices.

Draw the normal before the ray

Most Snell errors begin by measuring from the surface.

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Angles start at the normal

Draw the normal perpendicular to the boundary. Measure θ₁ and θ₂ between each ray and the normal, never the surface. Air has n ≈ 1.00, water n ≈ 1.33, and crown glass n ≈ 1.52.

An incident ray in air meets a crown-glass boundary. Both angles are measured from the dashed normal. The refracted ray bends toward the normal in higher-index crown glass. A dashed straight-line continuation shows where the ray would have gone without refraction. — Air is n ≈ 1.00 and crown glass is n ≈ 1.52. Measure both angles from the normal; the dashed line is an unrefracted reference.

Index and Snell equations

Refractive index

n=\frac{c}{v}

c = 3.00 × 10⁸ m/s; n has no unit.

Snell's law

n_1\sin\theta_1=n_2\sin\theta_2

Keep each index with its angle and use degree mode.

Unknown refracted angle

\theta_2=\sin^{-1}\!\left(\frac{n_1\sin\theta_1}{n_2}\right)

Isolate sine before inverse sine.

Predict the bend

Entering a higher-index material means slowing and bending toward the normal, so the angle decreases. Entering a lower-index material means speeding and bending away, so the angle increases. Frequency stays fixed because the source keeps its rate. Since v = fλ, wavelength changes with speed.

Snell calculation method

1. Label both media

Side 1 is incident; side 2 is transmitted.

2. Predict direction

Compare n₁ and n₂ before calculating.

3. Check degree mode

The source lab uses degrees.

4. Substitute and isolate

Keep n₁ with θ₁ and n₂ with θ₂.

5. Check the answer

The angle must be 0° to 90° and match the predicted bend.

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Worked examples

Air into glass

Air. $n_1=1.00$
Glass. $n_2=1.52$
Incident angle. $\theta_1=40.0^\circ$

Find θ₂.

Isolate sine.
$\sin\theta_2=\frac{(1.00)\sin40.0^\circ}{1.52}=0.4229$
Use inverse sine.
$\theta_2=25.0^\circ$

The smaller angle agrees with bending toward the normal.

Index from speed

Speed. $v=2.00\times10^8\,\text{m/s}$

Find n.

Divide c by v.
$n=\frac{3.00\times10^8}{2.00\times10^8}=1.50$

n = 1.50 with no unit.

Lab evidence

In Table 1, record the assigned indices and refraction angles, then show n₁ sin θ₁ and n₂ sin θ₂ are close within measurement limits. For Table 2, choose incidence angles of at least 30° and complete three trials for each material, using the results to calculate refractive index. The conclusion must state the purpose, describe the relationship between refractive index and refraction angle, and state what the analysis taught you. Do not present the worked example as a measured trial.

Refraction traps

Measuring from the surface

All Snell angles are measured from the normal.

Swapping labels

n₁ stays with the incident angle; n₂ stays with the refracted angle.

Radian mode

Use degree mode for degree-marked data.

Changing frequency

At a boundary, speed and wavelength change; source frequency remains fixed.

Snell's law practice

Predict toward or away before calculating.

Final checklist

Tick these off as you go. Saved in this browser.

I measure angles from the normal.
I label incident side 1 and transmitted side 2.
My calculator is in degree mode.
I completed Table 1 and showed both sides of Snell's law for the assigned rows.
My Table 2 angles are at least 30° and include three trials for each material.
My conclusion gives the purpose, the index-angle relationship, and what I learned from the data.
I reviewed the rubric and submitted 05.07 in Canvas.