Answer :
Sure, let's determine the index of refraction of the liquid given the angles of incidence and refraction. We will use Snell's Law to find the solution.
### Step-by-Step Solution
#### Step 1: Understand Snell's Law
Snell's Law describes the relationship between the angles of incidence and refraction when light passes through the boundary between two different media. It is represented mathematically as:
[tex]\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \][/tex]
Where:
- [tex]\( n_1 \)[/tex] is the index of refraction of the first medium (air in this case, typically [tex]\( n_1 = 1 \)[/tex])
- [tex]\( \theta_1 \)[/tex] is the angle of incidence
- [tex]\( n_2 \)[/tex] is the index of refraction of the second medium (the liquid we need to find)
- [tex]\( \theta_2 \)[/tex] is the angle of refraction
#### Step 2: Given Data
- Angle of incidence ([tex]\( \theta_1 \)[/tex]): 35 degrees
- Angle of refraction ([tex]\( \theta_2 \)[/tex]): 14 degrees
- Index of refraction for air ([tex]\( n_1 \)[/tex]): 1
#### Step 3: Convert Angles from Degrees to Radians
Since most trigonometric functions in mathematics use radians, we should convert the angles from degrees to radians.
The conversion formula is:
[tex]\[ \theta\, (\text{radians}) = \theta\, (\text{degrees}) \times \frac{\pi}{180} \][/tex]
For the angle of incidence:
[tex]\[ \theta_1 = 35^\circ \times \frac{\pi}{180} = \frac{35\pi}{180} \approx 0.6109 \, \text{radians} \][/tex]
For the angle of refraction:
[tex]\[ \theta_2 = 14^\circ \times \frac{\pi}{180} = \frac{14\pi}{180} \approx 0.2443 \, \text{radians} \][/tex]
#### Step 4: Apply Snell's Law
Using Snell's Law:
[tex]\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \][/tex]
Substitute [tex]\( n_1 \)[/tex], [tex]\( \theta_1 \)[/tex], and [tex]\( \theta_2 \)[/tex]:
[tex]\[ 1 \cdot \sin(0.6109) = n_2 \cdot \sin(0.2443) \][/tex]
#### Step 5: Calculate the Sine Values
[tex]\[ \sin(0.6109) \approx 0.5736 \][/tex]
[tex]\[ \sin(0.2443) \approx 0.2420 \][/tex]
#### Step 6: Solve for [tex]\( n_2 \)[/tex]
[tex]\[ 0.5736 = n_2 \cdot 0.2420 \][/tex]
[tex]\[ n_2 = \frac{0.5736}{0.2420} \approx 2.37 \][/tex]
### Answer
The index of refraction of the liquid is approximately 2.37.
### Step-by-Step Solution
#### Step 1: Understand Snell's Law
Snell's Law describes the relationship between the angles of incidence and refraction when light passes through the boundary between two different media. It is represented mathematically as:
[tex]\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \][/tex]
Where:
- [tex]\( n_1 \)[/tex] is the index of refraction of the first medium (air in this case, typically [tex]\( n_1 = 1 \)[/tex])
- [tex]\( \theta_1 \)[/tex] is the angle of incidence
- [tex]\( n_2 \)[/tex] is the index of refraction of the second medium (the liquid we need to find)
- [tex]\( \theta_2 \)[/tex] is the angle of refraction
#### Step 2: Given Data
- Angle of incidence ([tex]\( \theta_1 \)[/tex]): 35 degrees
- Angle of refraction ([tex]\( \theta_2 \)[/tex]): 14 degrees
- Index of refraction for air ([tex]\( n_1 \)[/tex]): 1
#### Step 3: Convert Angles from Degrees to Radians
Since most trigonometric functions in mathematics use radians, we should convert the angles from degrees to radians.
The conversion formula is:
[tex]\[ \theta\, (\text{radians}) = \theta\, (\text{degrees}) \times \frac{\pi}{180} \][/tex]
For the angle of incidence:
[tex]\[ \theta_1 = 35^\circ \times \frac{\pi}{180} = \frac{35\pi}{180} \approx 0.6109 \, \text{radians} \][/tex]
For the angle of refraction:
[tex]\[ \theta_2 = 14^\circ \times \frac{\pi}{180} = \frac{14\pi}{180} \approx 0.2443 \, \text{radians} \][/tex]
#### Step 4: Apply Snell's Law
Using Snell's Law:
[tex]\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \][/tex]
Substitute [tex]\( n_1 \)[/tex], [tex]\( \theta_1 \)[/tex], and [tex]\( \theta_2 \)[/tex]:
[tex]\[ 1 \cdot \sin(0.6109) = n_2 \cdot \sin(0.2443) \][/tex]
#### Step 5: Calculate the Sine Values
[tex]\[ \sin(0.6109) \approx 0.5736 \][/tex]
[tex]\[ \sin(0.2443) \approx 0.2420 \][/tex]
#### Step 6: Solve for [tex]\( n_2 \)[/tex]
[tex]\[ 0.5736 = n_2 \cdot 0.2420 \][/tex]
[tex]\[ n_2 = \frac{0.5736}{0.2420} \approx 2.37 \][/tex]
### Answer
The index of refraction of the liquid is approximately 2.37.
