Onc face of a rectangular glass plate 6 cm thick is silcerd. An object held 8 cm in front of the first face, froms an image 12 cm behind the silvered face. The refractive index of the glass is

Options

A. `0.4`
B. `0.8`
C. `1.2`
D. `1.6`

Answer

To solve the problem, we need to determine the refractive index of the glass plate based on the given distances and the behavior of light as it passes through the glass and reflects off the silvered surface.

### Step-by-Step Solution:

1. **Understand the Setup**:
- A rectangular glass plate is 6 cm thick.
- An object is placed 8 cm in front of the first face (let's call this face A).
- The second face (let's call this face B) is silvered, acting like a plane mirror.
- An image is formed 12 cm behind the silvered face (face B).

2. **Define Distances**:
- Let the distance from the object to face A be \( u = -8 \) cm (negative as per sign convention).
- Let the distance from the image to face B be \( v = +12 \) cm (positive as it is behind the mirror).

3. **Determine the Effective Object and Image Distances**:
- The distance from the object to the silvered face (face B) can be calculated as:
\[ \text{Distance from object to face B} = 8 \text{ cm} + 6 \text{ cm} = 14 \text{ cm} \]
- The distance from the image to face A (the first face) can be calculated as:
\[ \text{Distance from image to face A} = 12 \text{ cm} - 6 \text{ cm} = 6 \text{ cm} \]

4. **Use the Mirror Formula**:
- For a plane mirror, the relationship between object distance \( u \) and image distance \( v \) is given by:
\[ u + v = 0 \]
- However, we need to account for the thickness of the glass. The effective object distance from the silvered face is:
\[ u' = 8 + x \quad \text{(where \( x \) is the distance from face A to the image)} \]
- The effective image distance from the silvered face is:
\[ v' = 12 + 6 - x = 18 - x \]

5. **Set Up the Equation**:
- According to the plane mirror property:
\[ u' = v' \]
- Thus, we have:
\[ 8 + x = 18 - x \]
- Rearranging gives:
\[ 2x = 10 \implies x = 5 \text{ cm} \]

6. **Calculate the Refractive Index**:
- The real depth of the glass is the thickness, which is 6 cm.
- The apparent depth (which we found as \( x \)) is 5 cm.
- The refractive index \( \mu \) is given by:
\[ \mu = \frac{\text{Real Depth}}{\text{Apparent Depth}} = \frac{6}{5} = 1.2 \]

### Final Answer:
The refractive index of the glass is \( \mu = 1.2 \).