A ray of light is incident at an angle of `60^@` on one face of a rectangular glass slab of thickness `0.1 m`, and refractive glass slab of thickness `0.1 m`, and refractive index `1.5`. Calculate the lateral shift produced.

Explanation

To calculate the lateral shift produced by a ray of light incident on a rectangular glass slab, we can follow these steps:

### Step 1: Identify Given Values
- Angle of incidence (I) = 60°
- Thickness of the glass slab (t) = 0.1 m
- Refractive index of glass (n) = 1.5

### Step 2: Use Snell's Law to Find the Angle of Refraction (r)
Snell's Law states:
\[ n_1 \sin I = n_2 \sin r \]
Where:
- \( n_1 = 1 \) (refractive index of air)
- \( n_2 = 1.5 \) (refractive index of glass)
- \( I = 60° \)

Substituting the values:
\[ 1 \cdot \sin(60°) = 1.5 \cdot \sin(r) \]
\[ \sin(60°) = \frac{\sqrt{3}}{2} \]
Thus,
\[ \frac{\sqrt{3}}{2} = 1.5 \cdot \sin(r) \]
\[ \sin(r) = \frac{\sqrt{3}/2}{1.5} = \frac{\sqrt{3}}{3} \]
\[ \sin(r) = \frac{1}{\sqrt{3}} \]

Now, calculate \( r \):
\[ r = \arcsin\left(\frac{1}{\sqrt{3}}\right) \approx 35.3° \]

### Step 3: Calculate the Lateral Shift (D)
The formula for lateral shift (D) is given by:
\[ D = t \cdot \frac{\sin(I - r)}{\cos(r)} \]

Substituting the known values:
- \( t = 0.1 \) m
- \( I = 60° \)
- \( r \approx 35.3° \)

Calculate \( I - r \):
\[ I - r = 60° - 35.3° = 24.7° \]

Now substitute into the formula:
\[ D = 0.1 \cdot \frac{\sin(24.7°)}{\cos(35.3°)} \]

### Step 4: Calculate \( \sin(24.7°) \) and \( \cos(35.3°) \)
Using trigonometric values:
\[ \sin(24.7°) \approx 0.419 \]
\[ \cos(35.3°) \approx 0.819 \]

### Step 5: Substitute and Calculate D
Now substituting these values:
\[ D = 0.1 \cdot \frac{0.419}{0.819} \]
\[ D \approx 0.1 \cdot 0.511 \]
\[ D \approx 0.0511 \text{ m} \]

### Final Result
The lateral shift produced is approximately \( 0.0511 \text{ m} \) or \( 5.11 \text{ cm} \).

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