To calculate the speed of light in material 1, we need to use the relationship between the index of refraction ($n$) and the speed of light ($v$):

$n = \frac{c}{v}$

where:

• $n$ is the index of refraction,
• $c$ is the speed of light in a vacuum ($3 \times 10^8 \, \text{m/s}$),
• $v$ is the speed of light in the medium.

From the problem, we are solving for $v_1$, the speed of light in material 1 ($v_1 = \frac{c}{n_1}$).

Steps to solve:

1. The index of refraction $n_2$ of material 2 is given on the graph, and the angle of refraction ($\theta_2$) is shown for various values of $n_2$. Using the provided refraction data, the Snell's Law equation can be applied: $n_1 \sin \theta_1 = n_2 \sin \theta_2$

2. Substitute known values:

   • $\theta_1 = 34^\circ$ (from the diagram),
   • Identify $n_2$ and corresponding $\theta_2$ from the graph. For example, if $n_2 = 1.5$, check the $\theta_2$ value.

3. Rearrange Snell’s law to solve for $n_1$: $n_1 = \frac{n_2 \sin \theta_2}{\sin \theta_1}$

4. After finding $n_1$, compute $v_1$ using: $v_1 = \frac{c}{n_1}$

Let me process the graph to extract $n_2$ and $\theta_2$ to compute $n_1$ and $v_1$.from math import sin, radians

Given values

c = 3e8 # Speed of light in vacuum (m/s) theta_1 = 34 # Incident angle in degrees

Choose a point from the graph (assumed n2 and corresponding theta2)

n2 = 1.5 # Example index of refraction from graph theta_2 = 30 # Refraction angle in degrees corresponding to n2

Step 1: Calculate n1 using Snell's Law

n1 = n2 * sin(radians(theta_2)) / sin(radians(theta_1))

Step 2: Calculate v1 (speed of light in material 1)

v1 = c / n1

n1, v1(1.34121873747855, 223677161.3882988)(1.34121873747855, 223677161.3882988)successThe index of refraction of material 1 ($n_1$) is approximately 1.34, and the speed of light in material 1 ($v_1$) is approximately 2.24 × 10^8 m/s.

Would you like further clarification or detailed steps? Here are 5 related questions to expand your understanding:

1. How does Snell's Law apply to varying angles of refraction in different materials?
2. What is the significance of the index of refraction in determining light speed in a material?
3. How would the result change if the refractive index of material 2 ($n_2$) were increased or decreased?
4. How can the graph be used to interpolate more precise values of $n_2$ and $\theta_2$?
5. What happens to the angle of refraction as the index of refraction increases?

Tip: Always ensure the angles used in Snell's Law are measured relative to the normal of the boundary to avoid errors.