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Chapter 32: Problem 4

Standing by a pool filled with water, under what condition will you see a reflection of the scenery on the opposite side through total internal reflection of the light from the scenery? a) Your eyes are level with the water. b) You observe the pool at an angle of \(41.8^{\circ}\) c) Under no condition. d) You observe the pool at an angle of \(48.2^{\circ}\)

Short Answer

Expert verified
Answer: (c) Under no condition

Step by step solution

01

Determine the critical angle

First, we need to calculate the critical angle. We can do this using Snell's law: $$ n_{1} \times \sin(\theta_{1}) = n_{2} \times \sin(\theta_{2}) $$ where \(n_{1}\) and \(n_{2}\) are the refractive indices of the two media, and \(\theta_{1}\) and \(\theta_{2}\) are the angles of incidence and refraction, respectively. Here, \(n_{1}\) is the refractive index of water (approximately 1.33) and \(n_{2}\) is the refractive index of air (approximately 1). For total internal reflection to happen, \(\theta_{2}\) needs to be \(90^{\circ}\). Therefore, we can rewrite Snell's law as: $$ 1.33 \times \sin(\theta_{1c}) = 1 \times \sin(90^{\circ}) $$
02

Calculate the critical angle

Solve the equation in the previous step for \(\theta_{1c}\), the critical angle: $$ \sin(\theta_{1c}) = \frac{1}{1.33} $$ $$ \theta_{1c} = \sin^{-1} \left(\frac{1}{1.33}\right) $$ $$ \theta_{1c} \approx 48.6^{\circ} $$
03

Compare the critical angle to the given angles

Now that we have calculated the critical angle, let's assess each option to see if total internal reflection can occur: a) Your eyes are level with the water: In this case the incident angle is \(0^{\circ}\), which is less than the critical angle. So, total internal reflection will not occur. b) You observe the pool at an angle of \(41.8^{\circ}\): This angle is also less than the critical angle, so total internal reflection will not occur. c) Under no condition: We have not yet proven that this is true, so let's check the last option. d) You observe the pool at an angle of \(48.2^{\circ}\): This angle is slightly less than the critical angle, so total internal reflection will not occur.
04

Conclusion

Since none of the given angles led to total internal reflection, we can conclude that the answer is (c) Under no condition.

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Most popular questions from this chapter

Fermat's Principle, from which geometric optics can be derived, states that light travels by a path that minimizes the time of travel between the points. Consider a light beam that travels a horizontal distance \(D\) and a vertical distance \(h\), through two large flat slabs of material, with a vertical interface between the materials. One material has a thickness \(D / 2\) and index of refraction \(n_{1},\) and the second material has a thickness \(D / 2\) and index of refraction \(n_{2} .\) Determine the equation involving the indices of refraction and angles from horizontal that the light makes at the interface \(\left(\theta_{1}\right.\) and \(\theta_{2}\) ) which minimize the time for this travel.

You are using a mirror and a camera to make a self portrait. You focus the camera on yourself through the mirror. The mirror is a distance \(\mathrm{D}\) away from you. To what distance should you set the range of focus on the camera? a) \(D\) b) \(2 \mathrm{D}\) c) \(\mathrm{D} / 2\) d) \(4 \mathrm{D}\)

You have a spherical mirror with a radius of curvature of \(+20.0 \mathrm{~cm}\) (so it is concave facing you). You are looking at an object whose size you want to double in the image, so you can see it better. Where should you put the object? Where will the image be, and will it be real or virtual?

Answer as true or false with an explanation for the following: The wavelength of He-Ne laser light in water is less than its wavelength in the air. (The refractive index of water is \(1.33 .\)

Use Fermat's Principle to derive the law of reflection.
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