Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To determine the angle [tex]\(\theta_2\)[/tex] when a light wave travels from water into glass, we use Snell's Law, which can be written as:
[tex]\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \][/tex]
Here are the given values:
- [tex]\( n_1 = 1.33 \)[/tex] (refractive index of water)
- [tex]\( \theta_1 = 35^\circ \)[/tex] (angle in water)
- [tex]\( n_2 = 1.5 \)[/tex] (refractive index of glass)
We need to find [tex]\(\theta_2\)[/tex].
The formula to find [tex]\(\theta_2\)[/tex] is:
[tex]\[ \theta_2 = \sin^{-1}\left(\frac{n_1 \sin(\theta_1)}{n_2}\right) \][/tex]
Let's go through the steps in detail:
1. Convert the angle [tex]\(\theta_1\)[/tex] from degrees to radians because trigonometric functions in mathematical calculations usually require the angle in radians.
[tex]\[ \theta_1 = 35^\circ \][/tex]
Using the conversion factor [tex]\( \frac{\pi}{180} \)[/tex] to convert degrees to radians:
[tex]\[ \theta_1 \text{ in radians} = 35 \times \frac{\pi}{180} \approx 0.610865 \text{ radians} \][/tex]
2. Calculate [tex]\(\sin(\theta_1)\)[/tex] using the radian value obtained:
[tex]\[ \sin(0.610865) \approx 0.5736 \][/tex]
3. Apply Snell's Law to find [tex]\(\sin(\theta_2)\)[/tex]:
[tex]\[ \sin(\theta_2) = \frac{n_1 \sin(\theta_1)}{n_2} = \frac{1.33 \times 0.5736}{1.5} \approx 0.5087 \][/tex]
4. Determine [tex]\(\theta_2\)[/tex] by taking the inverse sine (arcsin) of the result:
[tex]\[ \theta_2 = \sin^{-1}(0.5087) \approx 0.533524 \text{ radians} \][/tex]
5. Convert [tex]\(\theta_2\)[/tex] back to degrees:
[tex]\[ \theta_2 \text{ in degrees} = 0.533524 \times \frac{180}{\pi} \approx 30.5687^\circ \][/tex]
6. Find the closest option to our calculated angle [tex]\(\theta_2\)[/tex] from the given multiple-choice answers:
- A. [tex]\(40.3^\circ\)[/tex]
- B. [tex]\(0.509^\circ\)[/tex]
- C. [tex]\(30.6^\circ\)[/tex]
- D. [tex]\(0.647^\circ\)[/tex]
Clearly, the value [tex]\(30.5687^\circ\)[/tex] is closest to option C: [tex]\(30.6^\circ\)[/tex].
Therefore, the correct answer is:
C. [tex]\(30.6^\circ\)[/tex]
[tex]\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \][/tex]
Here are the given values:
- [tex]\( n_1 = 1.33 \)[/tex] (refractive index of water)
- [tex]\( \theta_1 = 35^\circ \)[/tex] (angle in water)
- [tex]\( n_2 = 1.5 \)[/tex] (refractive index of glass)
We need to find [tex]\(\theta_2\)[/tex].
The formula to find [tex]\(\theta_2\)[/tex] is:
[tex]\[ \theta_2 = \sin^{-1}\left(\frac{n_1 \sin(\theta_1)}{n_2}\right) \][/tex]
Let's go through the steps in detail:
1. Convert the angle [tex]\(\theta_1\)[/tex] from degrees to radians because trigonometric functions in mathematical calculations usually require the angle in radians.
[tex]\[ \theta_1 = 35^\circ \][/tex]
Using the conversion factor [tex]\( \frac{\pi}{180} \)[/tex] to convert degrees to radians:
[tex]\[ \theta_1 \text{ in radians} = 35 \times \frac{\pi}{180} \approx 0.610865 \text{ radians} \][/tex]
2. Calculate [tex]\(\sin(\theta_1)\)[/tex] using the radian value obtained:
[tex]\[ \sin(0.610865) \approx 0.5736 \][/tex]
3. Apply Snell's Law to find [tex]\(\sin(\theta_2)\)[/tex]:
[tex]\[ \sin(\theta_2) = \frac{n_1 \sin(\theta_1)}{n_2} = \frac{1.33 \times 0.5736}{1.5} \approx 0.5087 \][/tex]
4. Determine [tex]\(\theta_2\)[/tex] by taking the inverse sine (arcsin) of the result:
[tex]\[ \theta_2 = \sin^{-1}(0.5087) \approx 0.533524 \text{ radians} \][/tex]
5. Convert [tex]\(\theta_2\)[/tex] back to degrees:
[tex]\[ \theta_2 \text{ in degrees} = 0.533524 \times \frac{180}{\pi} \approx 30.5687^\circ \][/tex]
6. Find the closest option to our calculated angle [tex]\(\theta_2\)[/tex] from the given multiple-choice answers:
- A. [tex]\(40.3^\circ\)[/tex]
- B. [tex]\(0.509^\circ\)[/tex]
- C. [tex]\(30.6^\circ\)[/tex]
- D. [tex]\(0.647^\circ\)[/tex]
Clearly, the value [tex]\(30.5687^\circ\)[/tex] is closest to option C: [tex]\(30.6^\circ\)[/tex].
Therefore, the correct answer is:
C. [tex]\(30.6^\circ\)[/tex]
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
