Snell’s Law

Side Project for Extra Credit

Estimate the refractive index of an unknown material from measured angles. When light moves from one medium to another, its path bends. This phenomenon is called refraction, and Snell’s law relates the angle of incidence $\theta_1$ to the angle of refraction $\theta_2$:

\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2), \]

where $n_1$ and $n_2$ are the refractive indices of the two media. Air has a refractive index close to 1, while water has a refractive index close to 1.33.

Main concepts: two-dimensional slicing, vectorized trigonometric functions, degree-to-radian conversion, and descriptive statistics.

The data

Each row contains an incidence angle followed by its corresponding refraction angle. Both are measured in degrees.

import numpy as np

# Columns: theta_1, theta_2
angles = np.array([
    [30, 18],
    [32, 19],
    [34, 21],
    [36, 22],
    [38, 23],
    [40, 24],
    [42, 25],
    [44, 26],
    [46, 27],
    [48, 28],
    [50, 29],
    [52, 30],
    [54, 31],
    [56, 32],
    [58, 33],
    [60, 33]
], dtype=float)

print(angles.shape)

(16, 2)

Our goal is to estimate $n_2$, the refractive index of the unknown material, given the index $n_1$ of the first medium. Every measurement gives one estimate, so we will summarize all of them with a mean and a standard deviation.

Step 1: Calculate relative refractive indices

Create index_of_refraction(theta_1, theta_2).

Accept two arrays of angles in degrees.
Convert both arrays to radians. NumPy’s trigonometric functions expect radians.
Return the array of ratios $\sin(\theta_1) / \sin(\theta_2)$.

Do not use loops: the function should work on whole arrays at once.

Step 2: Summarize the measurements

Create mean_and_std(values).

Calculate the mean and the population standard deviation of the array.
Return a string in the format mean +/- standard deviation, with both numbers displayed with two decimal places.

For [1, 1.41, 1.22], the result should be 1.21 +/- 0.17.

Step 3: Estimate the unknown index

Create estimate_index(angles_2d, n_1=1.0).

Extract the incidence and refraction columns from the two-dimensional array.
Call index_of_refraction().
Calculate $n_2 = n_1\sin(\theta_1)/\sin(\theta_2)$ for every measurement.
Call mean_and_std() and return the formatted estimate.

Step 4: Test and explain your estimate

Estimate the unknown material first with air as the initial medium and then with water, using n_1=1.33. Your results should be close to 1.58 +/- 0.02 and 2.10 +/- 0.02, respectively.

Then design small tests for:

Equal incidence and refraction angles, which should give a ratio of 1.
Your mean_and_std function with a list whose mean and standard deviation you can calculate by hand.
A single measurement, which should have a standard deviation of 0.00.

For each test, write the expected result before running your code. Explain why the angles must be converted to radians before calling np.sin(), and describe how measurement variability appears in the reported result.

Stretch goals (optional)

Plot $\sin(\theta_2)$ against $\sin(\theta_1)$. According to Snell’s law, the points should lie close to a straight line. Fit that line and compare its slope with your estimate.
The last measurement, [60, 33], looks slightly different from the rest. Estimate the index again without it and explain how much the result changes.

Before you submit

Upload your completed notebook to the Side Projects section on Blackboard before the final exam.

Include your functions, tests, saved outputs, and a short explanation of how measurement variability affects your estimate. Follow the notebook and AI-use guidance on the Side Projects page.

--- title: "Snell's Law" subtitle: "Side Project for Extra Credit" format: html --- Estimate the refractive index of an unknown material from measured angles. When light moves from one medium to another, its path bends. This phenomenon is called **refraction**, and Snell's law relates the angle of incidence $\theta_1$ to the angle of refraction $\theta_2$: $$ n_1 \sin(\theta_1) = n_2 \sin(\theta_2), $$ where $n_1$ and $n_2$ are the refractive indices of the two media. Air has a refractive index close to `1`, while water has a refractive index close to `1.33`. Main concepts: two-dimensional slicing, vectorized trigonometric functions, degree-to-radian conversion, and descriptive statistics. ## The data Each row contains an incidence angle followed by its corresponding refraction angle. Both are measured in degrees. ```{python} import numpy as np # Columns: theta_1, theta_2 angles = np.array([ [30, 18], [32, 19], [34, 21], [36, 22], [38, 23], [40, 24], [42, 25], [44, 26], [46, 27], [48, 28], [50, 29], [52, 30], [54, 31], [56, 32], [58, 33], [60, 33] ], dtype=float) print(angles.shape) ``` Our goal is to estimate $n_2$, the refractive index of the unknown material, given the index $n_1$ of the first medium. Every measurement gives one estimate, so we will summarize all of them with a mean and a standard deviation. ## Step 1: Calculate relative refractive indices Create `index_of_refraction(theta_1, theta_2)`. 1. Accept two arrays of angles in degrees. 2. Convert both arrays to radians. NumPy's trigonometric functions expect radians. 3. Return the array of ratios $\sin(\theta_1) / \sin(\theta_2)$. Do not use loops: the function should work on whole arrays at once. ## Step 2: Summarize the measurements Create `mean_and_std(values)`. 1. Calculate the mean and the population standard deviation of the array. 2. Return a string in the format `mean +/- standard deviation`, with both numbers displayed with two decimal places. For `[1, 1.41, 1.22]`, the result should be `1.21 +/- 0.17`. ## Step 3: Estimate the unknown index Create `estimate_index(angles_2d, n_1=1.0)`. 1. Extract the incidence and refraction columns from the two-dimensional array. 2. Call `index_of_refraction()`. 3. Calculate $n_2 = n_1\sin(\theta_1)/\sin(\theta_2)$ for every measurement. 4. Call `mean_and_std()` and return the formatted estimate. ## Step 4: Test and explain your estimate Estimate the unknown material first with air as the initial medium and then with water, using `n_1=1.33`. Your results should be close to `1.58 +/- 0.02` and `2.10 +/- 0.02`, respectively. Then design small tests for: - Equal incidence and refraction angles, which should give a ratio of `1`. - Your `mean_and_std` function with a list whose mean and standard deviation you can calculate by hand. - A single measurement, which should have a standard deviation of `0.00`. For each test, write the expected result before running your code. Explain why the angles must be converted to radians before calling `np.sin()`, and describe how measurement variability appears in the reported result. ## Stretch goals (optional) 1. Plot $\sin(\theta_2)$ against $\sin(\theta_1)$. According to Snell's law, the points should lie close to a straight line. Fit that line and compare its slope with your estimate. 2. The last measurement, `[60, 33]`, looks slightly different from the rest. Estimate the index again without it and explain how much the result changes. ## Before you submit Upload your completed notebook to the **Side Projects** section on Blackboard **before the final exam**. Include your functions, tests, saved outputs, and a short explanation of how measurement variability affects your estimate. Follow the notebook and AI-use guidance on the [Side Projects page](../structures/side_projects.qmd).