(II) For any type of wave that reaches a boundary beyond which its speed is increased, there is a maximum incident angle if there is to be a transmitted refracted wave. This maximum incident angle θ_iM corresponds to an angle of refraction equal to 90°. If θᵢ > θ_iM , all the wave is reflected at the boundary and none is refracted, because this would correspond to sin θᵣ > 1 (where is the angle θᵣ of refraction), which is impossible.
(a) Find a formula for θ_iM using the law of refraction, Eq. 15–19.
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Verified step by step guidance

1. Identify the law of refraction, also known as Snell's Law, which is given by the equation: \( n_1 \sin(\theta_i) = n_2 \sin(\theta_r) \), where \( n_1 \) and \( n_2 \) are the refractive indices of the two media, \( \theta_i \) is the incident angle, and \( \theta_r \) is the refracted angle.
2. Set the refracted angle \( \theta_r \) to 90°, because the problem states that the maximum incident angle corresponds to a refracted angle of 90°. This implies \( \sin(\theta_r) = \sin(90°) = 1 \).
3. Substitute \( \sin(\theta_r) = 1 \) into Snell's Law, resulting in the equation: \( n_1 \sin(\theta_i) = n_2 \).
4. Solve for \( \theta_i \) by isolating it on one side of the equation: \( \sin(\theta_i) = \frac{n_2}{n_1} \).
5. Find the maximum incident angle \( \theta_iM \) by taking the inverse sine (arcsin) of both sides: \( \theta_iM = \arcsin\left(\frac{n_2}{n_1}\right) \). This gives you the formula for the maximum incident angle when the refracted wave's angle is 90°.

Key Concepts

Law of Refraction (Snell's Law)

The Law of Refraction, also known as Snell's Law, describes how light or waves change direction when they pass from one medium to another. It is mathematically expressed as n₁ sin(θᵢ) = n₂ sin(θᵣ), where n₁ and n₂ are the refractive indices of the two media, and θᵢ and θᵣ are the angles of incidence and refraction, respectively. This law is fundamental in understanding how waves behave at boundaries between different materials.

Critical Angle

The critical angle is the angle of incidence above which total internal reflection occurs, meaning that all the incident wave is reflected back into the original medium. It is defined as θ_c = arcsin(n₂/n₁) when light travels from a medium with a higher refractive index (n₁) to one with a lower refractive index (n₂). Understanding the critical angle is essential for determining the maximum angle of incidence (θ_iM) for which refraction can still occur.

Total Internal Reflection

Total internal reflection is a phenomenon that occurs when a wave traveling in a medium hits a boundary with a lower refractive index at an angle greater than the critical angle. In this case, the wave cannot pass into the second medium and is instead completely reflected back. This concept is crucial for understanding why, at angles greater than θ_iM, no refraction occurs and all the wave energy is reflected.