One face of a rectangular glass plate 6 cm thick is silvered. An object held 8 cm in front of the first face, forms 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 step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the Setup
We have a rectangular glass plate that is 6 cm thick, with one face silvered (acting as a plane mirror). An object is placed 8 cm in front of the first face of the glass plate, and an image is formed 12 cm behind the silvered face.

### Step 2: Define Distances
- Let the thickness of the glass plate be \( t = 6 \) cm.
- The object distance from the first face of the glass plate is \( u = -8 \) cm (the negative sign indicates that the object is in front of the mirror).
- The image distance from the silvered face (acting as a mirror) is \( v = 12 \) cm (behind the mirror).

### Step 3: Relate Object and Image Distances
Since the silvered face acts as a plane mirror, the distance of the object from the mirror and the distance of the image from the mirror can be related as follows:

1. The distance of the object from the mirror is \( |u| + x \), where \( x \) is the apparent distance of the mirror from the observer's perspective.
2. The distance of the image from the mirror is \( |v| + (t - x) \), where \( t \) is the thickness of the glass.

Setting these equal because the image formed by a plane mirror is equal to the object distance, we have:

\[ |u| + x = |v| + (t - x) \]

### Step 4: Substitute Known Values
Substituting the known values into the equation:

\[ 8 + x = 12 + (6 - x) \]

### Step 5: Solve for \( x \)
Rearranging the equation gives:

\[ 8 + x = 12 + 6 - x \]
\[ 8 + x = 18 - x \]
\[ 2x = 18 - 8 \]
\[ 2x = 10 \]
\[ x = 5 \text{ cm} \]

### Step 6: Calculate the Refractive Index
Now that we have \( x \), we can find the refractive index \( \mu \) using the formula:

\[ \mu = \frac{\text{Real Depth}}{\text{Apparent Depth}} \]

The real depth is the thickness of the glass, which is \( 6 \) cm, and the apparent depth (the distance of the mirror from the observer's perspective) is \( 5 \) cm.

Substituting these values gives:

\[ \mu = \frac{6}{5} = 1.2 \]

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

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