(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 θiM corresponds to an angle of refraction equal to 90°. If θᵢ \u003e θiM, all the wave is reflected at the boundary and none is refracted, because this would correspond to sin θᵣ \u003e 1 (where is the angle θᵣ of refraction), which is impossible.(a) Find a formula for θiM using the law of refraction, Eq. 15–19.

Ch. 15 - Wave Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 15 - Wave Motion

Problem 66a

Chapter 15, Problem 66a

(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.

Verified step by step guidance

Step 1: Recall the law of refraction, also known as Snell's Law, which is expressed as:

n_{1} \sin θ i = n_{2} \sin θ r

. Here,

n_{1}

and

n_{2}

are the indices of refraction for the two media,

θ_{i}

is the angle of incidence, and

θ_{r}

is the angle of refraction.

Step 2: Recognize that the maximum incident angle

θ_{iM}

occurs when the angle of refraction

θ_{r}

is equal to 90°. Substitute

θ_{r}

= 90° into Snell's Law.

Step 3: Since

\sin (90)

is equal to 1, Snell's Law simplifies to:

n_{1} \sin θ_{iM} = n_{2}

. Rearrange this equation to solve for

θ_{iM}

Step 4: Divide both sides of the equation by

n_{1}

to isolate

\sin θ_{iM}

\sin θ_{iM} = \frac{n_{2}}{n_{1}}

Step 5: To find the maximum incident angle

θ_{iM}

, take the inverse sine (arcsin) of both sides:

θ_{iM} = \arcsin (\frac{n_{2}}{n_{1}})

. This is the formula for the maximum incident angle.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Law of Refraction (Snell's Law)

The Law of Refraction, also known as Snell's Law, describes how 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 light and other waves behave at boundaries.

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 n₁ > n₂. When the angle of incidence exceeds this critical angle, the refracted wave cannot exist, leading to the phenomenon described in the question.

Total Internal Reflection

Total internal reflection is a phenomenon that occurs when a wave traveling in a medium hits a boundary with a less dense medium at an angle greater than the critical angle. In this case, the wave is completely reflected back into the original medium, and no refraction occurs. This principle is crucial in applications such as fiber optics, where light is kept within the fiber by repeated total internal reflections.

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