For what angle of incidence, the lateral shift produced by a parallel sided glass plate is maximum ?

Options

A. `30^(@)`
B. `90^(@)`
C. `60^(@)`
D. `0^(@)`

Answer

To determine the angle of incidence at which the lateral shift produced by a parallel-sided glass plate is maximum, we can follow these steps:

### Step 1: Understand Lateral Shift
Lateral shift (S) is defined as the horizontal displacement of a light ray as it passes through a parallel-sided glass plate. It can be calculated using the formula:

\[ S = t \cdot \sec R \cdot \sin(I - R) \]

where:
- \( t \) = thickness of the glass plate
- \( I \) = angle of incidence
- \( R \) = angle of refraction

### Step 2: Analyze the Formula
To maximize the lateral shift \( S \), we need to maximize both \( \sec R \) and \( \sin(I - R) \).

### Step 3: Relate Angles Using Snell's Law
According to Snell's Law:

\[ n_1 \sin I = n_2 \sin R \]

where \( n_1 \) is the refractive index of air (approximately 1) and \( n_2 \) is the refractive index of glass.

### Step 4: Determine Maximum \( R \)
The angle of refraction \( R \) will be maximum when the angle of incidence \( I \) is also maximized. The maximum angle of incidence occurs when the light ray is incident parallel to the surface of the glass plate, which corresponds to the critical angle condition.

### Step 5: Identify the Critical Angle
The critical angle \( C \) can be defined as:

\[ C = \sin^{-1}\left(\frac{n_1}{n_2}\right) \]

For maximum lateral shift, we want \( I \) to approach 90 degrees, which means that \( R \) approaches the critical angle.

### Step 6: Conclusion
Thus, the angle of incidence \( I \) for which the lateral shift is maximum is:

\[ I = 90^\circ \]

### Final Answer
The angle of incidence at which the lateral shift produced by a parallel-sided glass plate is maximum is \( 90^\circ \).

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