One face of a rectangular glass plate `6 cm` thick is silvered. An object held `8 cm` in front of the unsilvered face forms an image `10 cm ` behind the silvered face.Find the refractive index of glass. Consider all the three steps.

Explanation

To find the refractive index of the glass plate, we can follow these steps:

### Step 1: Understand the arrangement
We have a rectangular glass plate that is 6 cm thick, with one face silvered (acting as a mirror). An object is placed 8 cm in front of the unsilvered face, and an image is formed 10 cm behind the silvered face.

### Step 2: Calculate the apparent distance of the object from the silvered face
When light passes through the glass, it refracts. The distance of the object from the silvered face (let's denote it as \( O' \)) can be calculated using the formula for apparent depth:
\[ O' = d + \mu \cdot h \]
where:
- \( d \) is the thickness of the glass (6 cm),
- \( h \) is the distance of the object from the unsilvered face (8 cm),
- \( \mu \) is the refractive index of glass.

Thus, the distance from the silvered face to the object is:
\[ O' = 6 + 8\mu \]

### Step 3: Set up the equation for the image distance
The image is formed 10 cm behind the silvered face. The total distance from the silvered face to the image is:
\[ I = O' + 6 = 6 + 8\mu + 6 = 12 + 8\mu \]

According to the problem, this distance is equal to the distance of the image from the silvered face, which is given as 10 cm. Therefore, we can set up the equation:
\[ 12 + 8\mu = 10\mu \]

### Step 4: Solve for the refractive index \( \mu \)
Rearranging the equation gives:
\[ 12 = 10\mu - 8\mu \]
\[ 12 = 2\mu \]
\[ \mu = \frac{12}{2} = 6 \]

### Step 5: Final answer
The refractive index of the glass is:
\[ \mu = \frac{3}{2} \]