The critical angle for refraction from glass to air is 30° and that from water to air is 37° . Find the critical angle for refraction from glass to water

Options

A. `sin^(-1)(5/6)`
B. `51^@3'`
C. `61^@2'`
D. `63^@3'`

Explanation

To find the critical angle for refraction from glass to water, we can use the information provided about the critical angles from glass to air and from water to air. Here’s a step-by-step solution:

### Step 1: Understand the Critical Angle
The critical angle (θc) is the angle of incidence beyond which light cannot pass through the boundary and is instead reflected back into the medium. It can be calculated using Snell's law.

### Step 2: Use Snell's Law for Glass to Air
For the transition from glass to air, we have:
- Critical angle (θc1) = 30°
- Refractive index of air (n_air) = 1
- Refractive index of glass (n_glass) = n_g

Using Snell's law:
\[ n_g \cdot \sin(30°) = n_{air} \cdot \sin(90°) \]
\[ n_g \cdot \frac{1}{2} = 1 \cdot 1 \]
\[ n_g = 2 \]

### Step 3: Use Snell's Law for Water to Air
For the transition from water to air, we have:
- Critical angle (θc2) = 37°
- Refractive index of water (n_water) = n_w

Using Snell's law:
\[ n_w \cdot \sin(37°) = n_{air} \cdot \sin(90°) \]
\[ n_w \cdot \sin(37°) = 1 \]
\[ n_w = \frac{1}{\sin(37°)} \]

Calculating \(\sin(37°)\):
\(\sin(37°) \approx 0.6018\)
So,
\[ n_w \approx \frac{1}{0.6018} \approx 1.66 \]

### Step 4: Find the Critical Angle from Glass to Water
Now, we can find the critical angle for the transition from glass to water using the formula:
\[ \sin(\theta_c) = \frac{n_w}{n_g} \]

Substituting the values we found:
\[ \sin(\theta_c) = \frac{n_w}{n_g} = \frac{1.66}{2} \]

Calculating:
\[ \sin(\theta_c) \approx 0.83 \]

### Step 5: Calculate the Critical Angle
Now, we find the critical angle by taking the inverse sine:
\[ \theta_c = \sin^{-1}(0.83) \]

Using a calculator:
\[ \theta_c \approx 56.44° \]

### Conclusion
The critical angle for refraction from glass to water is approximately **56.44°**.

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