A number of spherical conductors of different radii are charged to same potential. The surface charge density of each conductor is related with its radius as

Options

A. `sigmaprop(1)/(R^(2))`
B. `sigmaprop(1)/(R )`
C. `sigmapropR`
D. none of these

Answer

To solve the problem of how the surface charge density (σ) of spherical conductors relates to their radius (r) when they are charged to the same potential, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
- We have multiple spherical conductors with different radii (r1, r2, r3, ...).
- All these conductors are charged to the same potential (V).

2. **Using the Formula for Potential**:
- The potential (V) of a charged spherical conductor is given by the formula:
\[ V = \frac{KQ}{r} \]
where \( K \) is a constant (specifically \( \frac{1}{4\pi\epsilon_0} \)), \( Q \) is the charge on the conductor, and \( r \) is the radius.

3. **Setting Up the Equations**:
- Since all conductors are at the same potential, we can write:
\[ V_1 = V_2 = V_3 = ... = V_n \]
- This implies:
\[ \frac{KQ_1}{r_1} = \frac{KQ_2}{r_2} = \frac{KQ_3}{r_3} = ... = \frac{KQ_n}{r_n} \]
- From this, we can derive that:
\[ \frac{Q_1}{r_1} = \frac{Q_2}{r_2} = \frac{Q_3}{r_3} = ... = \frac{Q_n}{r_n} = C \]
where \( C \) is a constant.

4. **Expressing Charge in Terms of Radius**:
- From the above relation, we can express the charge \( Q \) in terms of the radius \( r \):
\[ Q = Cr \]

5. **Calculating Surface Charge Density**:
- The surface charge density \( \sigma \) is defined as:
\[ \sigma = \frac{Q}{A} \]
- For a sphere, the surface area \( A \) is given by:
\[ A = 4\pi r^2 \]
- Substituting \( Q = Cr \) into the equation for \( \sigma \):
\[ \sigma = \frac{Cr}{4\pi r^2} = \frac{C}{4\pi r} \]

6. **Determining the Relationship**:
- From the equation \( \sigma = \frac{C}{4\pi r} \), we can see that the surface charge density \( \sigma \) is inversely proportional to the radius \( r \):
\[ \sigma \propto \frac{1}{r} \]

### Conclusion:
Thus, the surface charge density of each conductor is inversely proportional to its radius.

### Final Answer:
The surface charge density \( \sigma \) is inversely proportional to the radius \( r \) of the spherical conductors.