A glass slab `(mu=1.5)` of thickness 6 cm is placed over a paper. What is the shift in the letters?

Options

A. `2` cm
B. `4/3` cm
C. `1/3` cm
D. `5/3` cm

Answer

To find the shift in the letters when a glass slab of thickness 6 cm and refractive index \( \mu = 1.5 \) is placed over a paper, we can follow these steps:

### Step 1: Understand the Geometry
We have a glass slab of thickness \( t = 6 \) cm placed over a paper. The light rays passing through the slab will experience refraction.

### Step 2: Identify the Shift
The shift in the letters is defined as the distance between the original position of the letters (the object) and the apparent position of the letters (the image) as seen through the glass slab.

### Step 3: Apply Snell's Law
Using Snell's law, we can relate the angles of incidence and refraction:
\[ \mu_1 \sin(i) = \mu_2 \sin(r) \]
Where:
- \( \mu_1 = 1.5 \) (refractive index of glass)
- \( \mu_2 = 1 \) (refractive index of air)
- \( i \) = angle of incidence
- \( r \) = angle of refraction

### Step 4: Approximate for Small Angles
For small angles, we can use the approximation:
\[ \sin(i) \approx \tan(i) \quad \text{and} \quad \sin(r) \approx \tan(r) \]
Thus, we can rewrite Snell's law as:
\[ 1.5 \tan(i) = 1 \tan(r) \]

### Step 5: Relate Tangents to Distances
In the triangle formed by the incident ray, we can express:
\[ \tan(i) = \frac{AB}{AO} \quad \text{and} \quad \tan(r) = \frac{AB}{AI} \]
Where:
- \( AB \) is the vertical distance from the original position of the letters to the point where the ray exits the slab.
- \( AO \) is the distance from the object to the point of incidence.
- \( AI \) is the distance from the object to the apparent position of the letters.

### Step 6: Set Up the Equation
From the previous equations, we can express:
\[ 1.5 \frac{AB}{AO} = \frac{AB}{AI} \]
Cancelling \( AB \) from both sides (assuming \( AB \neq 0 \)):
\[ 1.5 AO = AI \]

### Step 7: Solve for Distances
We know that the thickness of the slab \( t = AO \) is 6 cm. Therefore:
\[ AI = 1.5 \times AO = 1.5 \times 6 \text{ cm} = 9 \text{ cm} \]

### Step 8: Calculate the Shift
The shift \( S \) is given by:
\[ S = AO - AI \]
Substituting the values:
\[ S = 6 \text{ cm} - 9 \text{ cm} = -3 \text{ cm} \]
Since the shift is positive in the downward direction, we take the absolute value:
\[ S = 3 \text{ cm} \]

### Conclusion
The shift in the letters is 2 cm.