A light of wavelength 1 is incident on an object of size b. If a screen is at a distance D from the object, identify the correct condition for the obervation of different phenomena
a) if `b^(2) = D1`, Fresnel diffration is obserbed
b) if `b^(2) gt gt D1`, Fraunhofer diffraction is obserbed
c) if `b^(2) lt lt D1`, Fraunhofer diffraction is obserbed
d) if `b^(2) gt gt D1`, the approximation of geometrical optics is applicable

Options

A. a, b and d are true
B. a, c and d are true
C. a and c are true
D. a and d are true

Answer

To solve the problem, we need to analyze the conditions under which different types of diffraction phenomena occur based on the relationship between the size of the object (b), the distance to the screen (D), and the wavelength of light (λ).

### Step-by-Step Solution:

1. **Understanding the Parameters**:
- We have a light of wavelength \( \lambda = 1 \).
- The size of the object is \( b \).
- The distance from the object to the screen is \( D \).

2. **Fresnel Diffraction Condition**:
- Fresnel diffraction occurs when the size of the object squared \( b^2 \) is approximately equal to the distance \( D \) multiplied by the wavelength \( \lambda \).
- This can be expressed as:
\[ b^2 = D \cdot \lambda \]
- In this case, we can denote \( D \cdot \lambda \) as \( D_1 \). Therefore, the condition for Fresnel diffraction is:
\[ b^2 = D_1 \]

3. **Fraunhofer Diffraction Condition**:
- Fraunhofer diffraction occurs under different conditions. It is typically observed when the size of the object squared is much greater than the distance \( D \) multiplied by the wavelength \( \lambda \).
- This can be expressed as:
\[ b^2 \gg D \cdot \lambda \]
- Using \( D_1 \) again, this means:
\[ b^2 \gg D_1 \]

4. **Geometrical Optics Condition**:
- The approximation of geometrical optics is applicable when the size of the object squared is much greater than the distance to the screen multiplied by the wavelength:
\[ b^2 \gg D \cdot \lambda \]
- This is the same condition as for Fraunhofer diffraction.

5. **Summary of Conditions**:
- **Condition a**: \( b^2 = D_1 \) → Fresnel diffraction is observed (Correct).
- **Condition b**: \( b^2 \gg D_1 \) → Fraunhofer diffraction is observed (Correct).
- **Condition c**: \( b^2 \ll D_1 \) → Fraunhofer diffraction is observed (Incorrect; this condition is for Fresnel diffraction).
- **Condition d**: \( b^2 \gg D_1 \) → Geometrical optics is applicable (Correct).

### Final Answer:
The correct conditions for the observation of different phenomena are:
- a) Correct
- b) Correct
- c) Incorrect
- d) Correct