A number of spherical conductors of different radius have same potential. Then the surface charge density on them.

Options

A. is proportional to their radii
B. is inversely proportional to their radii
C. are equal
D. is proportional to square of their radii

Answer

To solve the problem of determining the surface charge density on spherical conductors of different radii that have the same potential, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept of Electric Potential**:
- For a spherical conductor, the electric potential \( V \) at its surface is given by the formula:
\[ V = \frac{kQ}{R} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge on the sphere, and \( R \) is the radius of the sphere.

2. **Setting Up the Equation for Different Spheres**:
- Let’s consider two spherical conductors with charges \( Q_1 \) and \( Q_2 \), and radii \( R_1 \) and \( R_2 \) respectively. Since they have the same potential:
\[ V_1 = V_2 \]
This implies:
\[ \frac{kQ_1}{R_1} = \frac{kQ_2}{R_2} \]

3. **Simplifying the Equation**:
- We can cancel \( k \) from both sides:
\[ \frac{Q_1}{R_1} = \frac{Q_2}{R_2} \]
- Rearranging gives us:
\[ Q_1 = Q_2 \cdot \frac{R_1}{R_2} \]

4. **Finding Surface Charge Density**:
- The surface charge density \( \sigma \) is defined as charge per unit area. For a sphere, the surface area \( A \) is given by:
\[ A = 4\pi R^2 \]
- Thus, the surface charge density for each sphere can be expressed as:
\[ \sigma_1 = \frac{Q_1}{4\pi R_1^2} \quad \text{and} \quad \sigma_2 = \frac{Q_2}{4\pi R_2^2} \]

5. **Substituting for Charge**:
- Using the relationship established earlier, we can substitute for \( Q_1 \):
\[ \sigma_1 = \frac{Q_2 \cdot \frac{R_1}{R_2}}{4\pi R_1^2} \]
- This simplifies to:
\[ \sigma_1 = \frac{Q_2}{4\pi R_2^2} \cdot \frac{1}{R_1} \]
- Similarly, for \( \sigma_2 \):
\[ \sigma_2 = \frac{Q_2}{4\pi R_2^2} \]

6. **Establishing the Relationship**:
- From the above equations, we can see that:
\[ \sigma_1 = \frac{\sigma_2}{R_1/R_2} \]
- This indicates that the surface charge density \( \sigma \) is inversely proportional to the radius \( R \):
\[ \sigma \propto \frac{1}{R} \]

### Conclusion:
Thus, we conclude that the surface charge density on the spherical conductors is inversely proportional to their radii.