A ray of light passes from glass, having a refractive indx of 1.6, to air. The angle of incidence for which the angle of refraction is twice the angle of incidence is

Options

A. `sin^(-1)((4)/(5))`
B. `sin^(-1)(3/(5))`
C. `sin^(-1)((5)/(8))`
D. `sin^(-1)((2)/(5))`

Explanation

To solve the problem, we need to find the angle of incidence (i) for which the angle of refraction (r) is twice the angle of incidence when light passes from glass to air. The refractive index of glass is given as 1.6.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We are given that the angle of refraction (r) is twice the angle of incidence (i). Therefore, we can write:
\[ r = 2i \]

2. **Applying Snell's Law**:
Snell's Law states that:
\[ n_1 \sin(i) = n_2 \sin(r) \]
Here, \( n_1 \) is the refractive index of glass (1.6), and \( n_2 \) is the refractive index of air (approximately 1). Thus, we can rewrite Snell's Law as:
\[ 1.6 \sin(i) = 1 \sin(r) \]

3. **Substituting for r**:
Since we know that \( r = 2i \), we can substitute this into Snell's Law:
\[ 1.6 \sin(i) = \sin(2i) \]

4. **Using the Double Angle Formula**:
The sine of double angle can be expressed as:
\[ \sin(2i) = 2 \sin(i) \cos(i) \]
Substituting this into the equation gives:
\[ 1.6 \sin(i) = 2 \sin(i) \cos(i) \]

5. **Simplifying the Equation**:
Assuming \( \sin(i) \neq 0 \), we can divide both sides by \( \sin(i) \):
\[ 1.6 = 2 \cos(i) \]
Rearranging gives:
\[ \cos(i) = \frac{1.6}{2} = 0.8 \]

6. **Finding the Angle of Incidence**:
Now, we can find the angle \( i \) using the inverse cosine function:
\[ i = \cos^{-1}(0.8) \]

7. **Calculating the Angle**:
Using a calculator, we find:
\[ i \approx 36.87^\circ \]

### Final Answer:
The angle of incidence \( i \) for which the angle of refraction is twice the angle of incidence is approximately:
\[ i \approx 37^\circ \]