Two concentric spheres of radii `R` and `2R` are charged. The inner sphere has a charge if `1muC` and the outer sphere has a charge of `2muC` of the same sigh. The potential is `9000 V` at a distance `3R` from the common centre. The value of R is

Options

A. `1m`
B. `2m`
C. `3m`
D. `4m`

Explanation

To solve the problem, we need to calculate the value of \( R \) given the potential at a distance of \( 3R \) from the center of two concentric spheres with known charges.

### Step-by-Step Solution:

1. **Identify the Charges and Distances:**
- The inner sphere has a charge \( q_1 = 1 \, \mu C = 1 \times 10^{-6} \, C \).
- The outer sphere has a charge \( q_2 = 2 \, \mu C = 2 \times 10^{-6} \, C \).
- The distance from the center where the potential is measured is \( r = 3R \).

2. **Formula for Electric Potential:**
The electric potential \( V \) at a distance \( r \) from a point charge \( q \) is given by:
\[ V = k \frac{q}{r} \]
where \( k \) is Coulomb's constant, \( k \approx 9 \times 10^9 \, N \cdot m^2/C^2 \).

3. **Calculate the Total Potential at Distance \( 3R \):**
The total potential \( V \) at a distance \( 3R \) due to both spheres is the sum of the potentials due to each sphere:
\[ V = V_1 + V_2 = k \frac{q_1}{3R} + k \frac{q_2}{3R} \]
Factoring out \( k/3R \):
\[ V = \frac{k}{3R} (q_1 + q_2) \]

4. **Substituting the Values:**
Substitute \( q_1 \) and \( q_2 \):
\[ V = \frac{k}{3R} (1 \times 10^{-6} + 2 \times 10^{-6}) = \frac{k}{3R} (3 \times 10^{-6}) \]

5. **Set the Potential Equal to the Given Value:**
We know from the problem statement that the potential at \( 3R \) is \( 9000 \, V \):
\[ \frac{k}{3R} (3 \times 10^{-6}) = 9000 \]

6. **Rearranging the Equation:**
Rearranging the equation to solve for \( R \):
\[ R = \frac{k \cdot 3 \times 10^{-6}}{3 \cdot 9000} \]
Simplifying:
\[ R = \frac{k \cdot 10^{-6}}{9000} \]

7. **Substituting the Value of \( k \):**
Substitute \( k = 9 \times 10^9 \):
\[ R = \frac{(9 \times 10^9) \cdot (10^{-6})}{9000} \]
Simplifying further:
\[ R = \frac{9 \times 10^3}{9000} = 1 \, m \]

### Final Answer:
The value of \( R \) is \( 1 \, m \).