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 \(\(30^\circ\)\), we can use Snell's Law. Here is the step-by-step solution:
### Step 1: Identify the known variables
- Refractive index of air (\(\(n_1\)\)): 1
- Refractive index of water (\(\(n_2\)\)): 1.32
- Angle of refraction (\(\(\theta_2\)\)): 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.
\(\[
\theta_2 = 30^\circ
\]\)
To convert degrees to radians, use the conversion factor \(\(\frac{\pi}{180}\)\):
\(\[
\theta_2 \, \text{(in radians)} = 30^\circ \times \frac{\pi}{180} = \frac{\pi}{6} \approx 0.5236 \, \text{radians}
\]\)
### Step 3: Apply Snell's Law
Snell's Law relates the angles and the indices of refraction of the two mediums:
\(\[
n_1 \sin(\theta_1) = n_2 \sin(\theta_2)
\]\)
Where:
- \(\(n_1\)\) is the refractive index of the first medium (air).
- \(\(\theta_1\)\) is the angle of incidence.
- \(\(n_2\)\) is the refractive index of the second medium (water).
- \(\(\theta_2\)\) is the angle of refraction.
### Step 4: Solve for \(\(\sin(\theta_1)\)\)
We need to find \(\(\sin(\theta_1)\)\):
\(\[
\sin(\theta_1) = \frac{n_2 \sin(\theta_2)}{n_1}
\]\)
Substitute the known values into the equation:
\(\[
\sin(\theta_1) = \frac{1.32 \sin(0.5236)}{1}
\]\)
Using the value of \(\(\sin(30^\circ) = 0.5\)\):
\(\[
\sin(\theta_1) = 1.32 \times 0.5 = 0.66
\]\)
### Step 5: Find the angle of incidence \(\(\theta_1\)\)
Now that we have \(\(\sin(\theta_1) = 0.66\)\), we need to find \(\(\theta_1\)\):
\(\[
\theta_1 = \arcsin(0.66)
\]\)
Converting the result from radians to degrees:
\(\[
\theta_1 \approx 41.3^\circ
\]\)
### 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 \(\(30^\circ\)\), is approximately \(\(41.3^\circ\)\).
