To determine the angle of incidence of a light ray incident through air on the boundary separating air from water such that the angle of refraction is [tex]\(30^\circ\)[/tex], we can use Snell's Law. Here is the step-by-step solution:
### Step 1: Identify the known variables
- Refractive index of air ([tex]\(n_1\)[/tex]): 1
- Refractive index of water ([tex]\(n_2\)[/tex]): 1.32
- Angle of refraction ([tex]\(\theta_2\)[/tex]): 30°
### Step 2: Convert the angle of refraction from degrees to radians
We need to convert the angle of refraction from degrees to radians because trigonometric functions in higher-level mathematics typically require angles in radians.
[tex]\[
\theta_2 = 30^\circ
\][/tex]
To convert degrees to radians, use the conversion factor [tex]\(\frac{\pi}{180}\)[/tex]:
[tex]\[
\theta_2 \, \text{(in radians)} = 30^\circ \times \frac{\pi}{180} = \frac{\pi}{6} \approx 0.5236 \, \text{radians}
\][/tex]
### Step 3: Apply Snell's Law
Snell's Law relates the angles and the indices of refraction of the two mediums:
[tex]\[
n_1 \sin(\theta_1) = n_2 \sin(\theta_2)
\][/tex]
Where:
- [tex]\(n_1\)[/tex] is the refractive index of the first medium (air).
- [tex]\(\theta_1\)[/tex] is the angle of incidence.
- [tex]\(n_2\)[/tex] is the refractive index of the second medium (water).
- [tex]\(\theta_2\)[/tex] is the angle of refraction.
### Step 4: Solve for [tex]\(\sin(\theta_1)\)[/tex]
We need to find [tex]\(\sin(\theta_1)\)[/tex]:
[tex]\[
\sin(\theta_1) = \frac{n_2 \sin(\theta_2)}{n_1}
\][/tex]
Substitute the known values into the equation:
[tex]\[
\sin(\theta_1) = \frac{1.32 \sin(0.5236)}{1}
\][/tex]
Using the value of [tex]\(\sin(30^\circ) = 0.5\)[/tex]:
[tex]\[
\sin(\theta_1) = 1.32 \times 0.5 = 0.66
\][/tex]
### Step 5: Find the angle of incidence [tex]\(\theta_1\)[/tex]
Now that we have [tex]\(\sin(\theta_1) = 0.66\)[/tex], we need to find [tex]\(\theta_1\)[/tex]:
[tex]\[
\theta_1 = \arcsin(0.66)
\][/tex]
Converting the result from radians to degrees:
[tex]\[
\theta_1 \approx 41.3^\circ
\][/tex]
### Final Answer
The angle of incidence of the light ray incident through air on the boundary separating air from water, so that the angle of refraction is [tex]\(30^\circ\)[/tex], is approximately [tex]\(41.3^\circ\)[/tex].
