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 a light ray passing through a glass slab of thickness \( t \) with an angle of incidence \( i \) and an angle of refraction \( r \), we can derive the expression step by step.

### Step 1: Understand the Geometry
When light enters a glass slab, it bends at the interface due to refraction. The angle of incidence \( i \) and the angle of refraction \( r \) are related by 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 2: Identify the Lateral Shift
The lateral shift \( s \) is the horizontal distance the light ray is displaced as it passes through the slab. We can analyze the triangles formed by the incident ray, refracted ray, and the thickness of the slab.

### Step 3: Use Trigonometric Relationships
In the triangle formed by the incident ray and the refracted ray:
- The angle at the top of the triangle is \( i - r \).
- The thickness of the slab is \( t \).

Using the sine function:
\[ \sin(i - r) = \frac{s}{AB} \]
where \( AB \) is the distance traveled by the ray within the slab.

### Step 4: Relate the Thickness to the Base
In the triangle formed by the thickness \( t \) and the angle \( r \):
\[ \cos(r) = \frac{t}{AB} \]
From this, we can express \( AB \):
\[ AB = \frac{t}{\cos(r)} \]

### Step 5: Substitute \( AB \) in the Lateral Shift Equation
Now substituting \( AB \) back into the lateral shift equation:
\[ \sin(i - r) = \frac{s}{\frac{t}{\cos(r)}} \]
This simplifies to:
\[ s = t \cdot \sin(i - r) \cdot \cos(r) \]

### Step 6: Final Expression for Lateral Shift
Thus, the lateral shift \( s \) can be expressed as:
\[ s = t \cdot \frac{\sin(i - r)}{\cos(r)} \]

### Conclusion
The final expression for the lateral shift \( s \) of the light ray as it passes through the glass slab is:
\[ s = t \cdot \frac{\sin(i - r)}{\cos(r)} \]