In the above question, the surface charge densities have the correct relation is-

A. `sigma_(A) gt sigma_(B) gt sigma_(C) gt sigma_(D)`
B. `sigma_(A)=sigma_(B)=sigma_(C)=sigma_(D)`
C. `sigma_(D) gt sigma_(C) gt sigma_(B) gt sigma_(A)`
D. `sigma_(C) lt sigma_(B) gt sigma_(A) gt sigma_(D)`

Options

A. `sigma_(A) gt sigma_(B) gt sigma_(C) gt sigma_(D)`
B. `sigma_(A)=sigma_(B)=sigma_(C)=sigma_(D)`
C. `sigma_(D) gt sigma_(C) gt sigma_(B) gt sigma_(A)`
D. `sigma_(C) lt sigma_(B) gt sigma_(A) gt sigma_(D)`

Answer

To solve the problem regarding the surface charge densities on different parts of a conductor, we will analyze the relationship based on the geometry of the surfaces involved.

### Step-by-Step Solution:

1. **Identify the Surfaces**:
We have four surfaces on the conductor:
- Surface A: Pointed end
- Surface B: Small semicircle
- Surface C: Long semicircle
- Surface D: Flat surface

2. **Understand the Geometry**:
- The radius of curvature for each surface is different:
- Surface A (pointed) has the smallest radius of curvature.
- Surface B (small semicircle) has a larger radius than A but smaller than C.
- Surface C (long semicircle) has a larger radius than B.
- Surface D (flat surface) has an infinite radius of curvature.

3. **Apply the Concept of Charge Density**:
- The surface charge density (σ) is inversely proportional to the radius of curvature (R). This means:
\[ \sigma \propto \frac{1}{R} \]
- Therefore, the smaller the radius of curvature, the higher the surface charge density.

4. **Establish the Relationships**:
- Since Surface A has the smallest radius, it will have the highest charge density:
\[ \sigma_A > \sigma_B > \sigma_C > \sigma_D \]
- This implies:
- \(\sigma_A\) (pointed end) > \(\sigma_B\) (small semicircle) > \(\sigma_C\) (long semicircle) > \(\sigma_D\) (flat surface).

5. **Select the Correct Option**:
- Based on the relationships derived, we can conclude that the correct option is:
- **A.** \(\sigma_A > \sigma_B > \sigma_C > \sigma_D\)

### Final Answer:
The correct relation of the surface charge densities is:
**A.** \(\sigma_A > \sigma_B > \sigma_C > \sigma_D\)