Bending Light and Snell's Law

Light bends! Light travels at different speeds through different substances and changes direction as it changes speed. A pole lowered into a pool of water appears to bend. A man standing in waist-deep water appears shorter. 

In this Pull-Out Lesson, we will discuss the mathematics behind the phenomenon of bending light. We will look at Snell's Law which relates the size of the bend to the change in speed as light passes from one substance to another. The student will be able to see why a window does not appear to bend the scenery even though light travels slower in glass than in air. Does sound bend also?

Note: The information below was created with the assistance of AI.

Level of Mathematics

This module is suitable for:

• High school students, primarily grades 10–12, especially those taking Algebra II, Precalculus, or Introductory Physics.

• It may also be appropriate for college preparatory students in an interdisciplinary math/physics class.

• The content includes trigonometric functions (sine, inverse sine), use of calculators, and simple algebraic manipulation, but no calculus is required for the core lesson (though a reference to calculus-based derivation is made).

Application Areas

The lesson connects mathematics with:

• Physics: Specifically, the refraction of light and Snell's Law.

• Optics: Explaining visual phenomena like distorted images through water or glass.

• Everyday Observations: Bent poles in water, why legs look shorter in a swimming pool, why windows don’t distort scenery.

• Geoscience: Mentioned in extension (e.g., how bending sound waves are used in studying Earth’s structure).

These connections help students see the real-world relevance of mathematical relationships.

Prerequisites

To successfully engage with the material, students should be comfortable with:

• Basic trigonometry: Understanding and applying the sine function.

• Using scientific calculators: Including evaluating sine and inverse sine.

• Algebraic manipulation: Rearranging formulas and interpreting ratios.

• Angle measurement and geometry: Using protractors, understanding perpendicular lines, angles of incidence and refraction.

Some prior exposure to concepts in physical science or physics would enhance comprehension but is not strictly necessary.

Subject Matter

The mathematical concepts and content covered include:

• Snell’s Law: sin⁡asin⁡β=v1v2\frac{\sin a}{\sin \beta} = \frac{v_1}{v_2}sinβsina​=v2​v1​​, where v1v_1v1​ and v2v_2v2​ are the speeds of light in two media, and $a$, $\beta$ are angles of incidence and refraction.

• Trigonometric analysis: Use of sine and inverse sine to model light refraction.

• Critical angles and total internal reflection: Explored through error conditions on calculators.

• Angle relationships: Measurement from the perpendicular, and symmetry in double refractions (e.g., entering and exiting through a windowpane).

• Visual distortions: Applications in explaining common phenomena using geometric optics.

Students also engage in problem-solving, experimentation, and reasoning from diagrams.

Correlation to Mathematics Standards

Common Core Math Standards – High School

Functions – Trigonometric Functions

• HSF-TF.A.1: Understand radian and degree measures of angles.

• HSF-TF.A.3: Use special triangles to determine trigonometric ratios.

• HSF-TF.C.8: Prove and apply trigonometric identities.

Geometry

• HSG-SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles.

• HSG-MG.A.1–3: Apply geometric concepts in modeling situations (e.g., light bending across mediums).

Mathematical Practices

• MP4: Model with mathematics – using a real-world physical phenomenon to apply trigonometry.

• MP5: Use appropriate tools strategically – including calculators and protractors.

• MP6: Attend to precision – especially when interpreting trigonometric relationships and angle measures.

• MP2: Reason abstractly and quantitatively – by interpreting speed ratios and sine values.