A charge Q is uniformly distributed over the surface of two conducting concentric spheres of radii R and r (Rgtr). Then, potential at common centre of these spheres is

Options

A. `(KQ(R+r))/(Rr)`
B. `(KQ(R+r))/((R^2+r^2))`
C. `(KQ)/(sqrt(R^2+r^2))`
D. `KQ(1/R-1/r)`

Answer

To find the potential at the common center of two concentric conducting spheres with radii \( R \) and \( r \) (where \( R > r \)), we can follow these steps:

### Step 1: Understand the Charge Distribution
We have a total charge \( Q \) uniformly distributed over the surfaces of both spheres. Let:
- \( Q_1 \) be the charge on the inner sphere (radius \( r \))
- \( Q_2 \) be the charge on the outer sphere (radius \( R \))

We know that:
\[ Q_1 + Q_2 = Q \]

### Step 2: Calculate the Surface Charge Densities
The surface charge density \( \sigma \) for both spheres can be defined as:
- For the inner sphere:
\[ \sigma_1 = \frac{Q_1}{4\pi r^2} \]
- For the outer sphere:
\[ \sigma_2 = \frac{Q_2}{4\pi R^2} \]

### Step 3: Express Charges in Terms of Charge Densities
From the surface charge densities, we can express \( Q_1 \) and \( Q_2 \):
- \( Q_1 = \sigma_1 \cdot 4\pi r^2 \)
- \( Q_2 = \sigma_2 \cdot 4\pi R^2 \)

### Step 4: Write the Potential at the Center
The potential \( V \) at the common center \( O \) due to a charged sphere is given by:
- For the inner sphere:
\[ V_1 = \frac{k Q_1}{r} \]
- For the outer sphere:
\[ V_2 = \frac{k Q_2}{R} \]

Thus, the total potential at point \( O \) is:
\[ V_O = V_1 + V_2 = \frac{k Q_1}{r} + \frac{k Q_2}{R} \]

### Step 5: Substitute \( Q_1 \) and \( Q_2 \) in Terms of \( Q \)
Using \( Q_1 + Q_2 = Q \), we can express \( V_O \) in terms of \( Q \):
\[ V_O = k \left( \frac{Q_1}{r} + \frac{Q_2}{R} \right) \]

### Step 6: Find \( Q_1 \) and \( Q_2 \) in Terms of \( Q \)
From the charge density equations:
\[ Q_1 = \sigma_1 \cdot 4\pi r^2 = \frac{Q}{4\pi (r^2 + R^2)} \cdot 4\pi r^2 = \frac{Qr^2}{r^2 + R^2} \]
\[ Q_2 = \sigma_2 \cdot 4\pi R^2 = \frac{Q}{4\pi (r^2 + R^2)} \cdot 4\pi R^2 = \frac{QR^2}{r^2 + R^2} \]

### Step 7: Substitute Back into the Potential Equation
Now substituting \( Q_1 \) and \( Q_2 \) into the potential equation:
\[ V_O = k \left( \frac{Qr^2}{r(r^2 + R^2)} + \frac{QR^2}{R(r^2 + R^2)} \right) \]

### Step 8: Simplify the Expression
This simplifies to:
\[ V_O = kQ \left( \frac{r + R}{r^2 + R^2} \right) \]

### Final Answer
Thus, the potential at the common center of the spheres is:
\[ V_O = \frac{kQ(r + R)}{r^2 + R^2} \]