Answer :
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]
