A light enters a glass slab of thickness `t` with angle of incidence `i` suffers lateral shifts(s).If angle of refraction of light is `r` , then lateral shift is equal to

Options

A. `(t × cos(i – r))/sin i`
B. `(t × sin(i – r))/cos r`
C. `(t × sin(i – r))/cos i`
D. `(t × cos(i – r))/cos r`

Answer

To find the lateral shift \( S \) of light entering a glass slab of thickness \( t \) at an angle of incidence \( i \) and an angle of refraction \( r \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- When light enters the glass slab, it refracts at an angle \( r \). The lateral shift \( S \) can be visualized as the horizontal distance the light ray shifts while passing through the slab.

2. **Identifying the Angles**:
- The angle of incidence is \( i \).
- The angle of refraction is \( r \).
- The angle between the incident ray and the normal is \( i \), and the angle between the refracted ray and the normal is \( r \).

3. **Using Trigonometry**:
- In the triangle formed by the incident ray, the refracted ray, and the lateral shift, we can use the sine function to relate the angles and the sides.
- From the triangle \( ABM' \) (where \( A \) is the point of incidence, \( B \) is the point where the ray exits, and \( M' \) is the point of lateral shift), we have:
\[ \sin(i - r) = \frac{S}{AB} \]
- Here, \( S \) is the lateral shift and \( AB \) is the hypotenuse.

4. **Finding the Length of \( AB \)**:
- In triangle \( ABM \) (where \( M \) is the thickness of the slab), we can relate the thickness \( t \) of the slab to \( AB \) using the cosine function:
\[ \cos(r) = \frac{AM}{AB} \]
- Rearranging gives:
\[ AB = \frac{t}{\cos(r)} \]

5. **Substituting \( AB \) into the Sine Equation**:
- Substitute \( AB \) back into the sine equation:
\[ \sin(i - r) = \frac{S}{\frac{t}{\cos(r)}} \]
- This simplifies to:
\[ S = t \cdot \sin(i - r) \cdot \cos(r) \]

6. **Final Expression for Lateral Shift**:
- Therefore, the lateral shift \( S \) can be expressed as:
\[ S = t \cdot \frac{\sin(i - r)}{\cos(r)} \]

### Final Formula:
\[ S = t \cdot \sin(i - r) \cdot \sec(r) \]