Two concentric, thin metallic spheres of radii `R_(1) and R_(2) (R_(1) gt R_(2))` charges `Q_(1) and Q_(2)` respectively Then the potential at radius r between `R_(1) and R_(2)` will be `(k = 1//4pi in)`

Options

A. A) `k((Q_(1) + Q_(2))/(r))`
B. B) `K((Q_(1))/(r)+(Q_(2))/(R_(2)))`
C. `k((Q_(2))/(r)+(Q_(1))/(R_(1)))`
D. D) `k((Q_(1))/(R_(1))+(Q_(2))/(R_(2)))`

Answer

To find the potential at a radius \( r \) between two concentric metallic spheres with radii \( R_1 \) and \( R_2 \) (where \( R_1 > R_2 \)) and charges \( Q_1 \) and \( Q_2 \) respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have two concentric metallic spheres. The outer sphere has a radius \( R_1 \) and charge \( Q_1 \), while the inner sphere has a radius \( R_2 \) and charge \( Q_2 \). The region of interest is between the two spheres, where \( R_2 < r < R_1 \).

2. **Potential Due to a Charged Sphere**:
- The electric potential \( V \) due to a charged sphere at a distance \( r \) from its center (for \( r \) greater than the radius of the sphere) is given by:
\[ V = k \frac{Q}{r} \]
- Here, \( k = \frac{1}{4 \pi \epsilon_0} \).

3. **Potential in the Region Between the Spheres**:
- The potential at a distance \( r \) from the center due to the inner sphere (radius \( R_2 \) and charge \( Q_2 \)) is:
\[ V_2 = k \frac{Q_2}{r} \]
- The potential due to the outer sphere (radius \( R_1 \) and charge \( Q_1 \)) is constant throughout the region outside the sphere and is given by:
\[ V_1 = k \frac{Q_1}{R_1} \]
- Since we are in the region between the two spheres, the potential due to the outer sphere \( V_1 \) remains constant.

4. **Total Potential**:
- The total potential \( V \) at a distance \( r \) between the two spheres is the sum of the potentials from both spheres:
\[ V = V_1 + V_2 = k \frac{Q_1}{R_1} + k \frac{Q_2}{r} \]
- We can factor out \( k \):
\[ V = k \left( \frac{Q_1}{R_1} + \frac{Q_2}{r} \right) \]

5. **Final Expression**:
- Thus, the potential at radius \( r \) between the two spheres is:
\[ V = k \left( \frac{Q_1}{R_1} + \frac{Q_2}{r} \right) \]

### Conclusion:
The potential at radius \( r \) between the two concentric spheres is given by:
\[ V = k \left( \frac{Q_1}{R_1} + \frac{Q_2}{r} \right) \]