A charge Q is distributed over two concentric hollow spheres of radii r and `R (gt r)` such that the surface charge densities are equal. Find the potential at the common centre.

Explanation

To find the potential at the common center of two concentric hollow spheres with equal surface charge densities, we can follow these steps:

### Step 1: Understand the Problem
We have two concentric hollow spheres with radii \( r \) (inner sphere) and \( R \) (outer sphere), where \( R > r \). The total charge \( Q \) is distributed over both spheres such that their surface charge densities are equal.

### Step 2: Define Surface Charge Density
The surface charge density \( \sigma \) for a sphere is defined as:
\[ \sigma = \frac{Q}{A} \]
where \( A \) is the surface area of the sphere. For the inner sphere, the area is \( 4\pi r^2 \) and for the outer sphere, the area is \( 4\pi R^2 \).

### Step 3: Set Up the Equations for Surface Charge Densities
Let \( Q_1 \) be the charge on the inner sphere and \( Q_2 \) be the charge on the outer sphere. Since the surface charge densities are equal, we have:
\[ \sigma_1 = \sigma_2 \]
This gives us:
\[ \frac{Q_1}{4\pi r^2} = \frac{Q_2}{4\pi R^2} \]
From this, we can express \( Q_2 \) in terms of \( Q_1 \):
\[ Q_2 = Q_1 \cdot \frac{R^2}{r^2} \]

### Step 4: Relate Charges to Total Charge
Since the total charge \( Q \) is distributed over both spheres, we have:
\[ Q = Q_1 + Q_2 \]
Substituting \( Q_2 \) from the previous step:
\[ Q = Q_1 + Q_1 \cdot \frac{R^2}{r^2} \]
Factoring out \( Q_1 \):
\[ Q = Q_1 \left(1 + \frac{R^2}{r^2}\right) \]
Solving for \( Q_1 \):
\[ Q_1 = \frac{Q}{1 + \frac{R^2}{r^2}} = \frac{Qr^2}{r^2 + R^2} \]

### Step 5: Calculate the Potential at the Center
The potential \( V \) at a point inside a charged spherical shell (like our inner sphere) is given by:
\[ V = \frac{KQ}{R} \]
For the inner sphere:
\[ V_1 = \frac{KQ_1}{r} \]
For the outer sphere (since it is outside the inner sphere):
\[ V_2 = \frac{KQ_2}{R} \]
Substituting \( Q_2 \):
\[ V_2 = \frac{KQ_1 \cdot \frac{R^2}{r^2}}{R} = \frac{KQ_1 R}{r^2} \]

### Step 6: Total Potential at the Center
The total potential at the center due to both spheres is:
\[ V_{\text{total}} = V_1 + V_2 = \frac{KQ_1}{r} + \frac{KQ_1 R}{r^2} \]
Factoring out \( KQ_1 \):
\[ V_{\text{total}} = KQ_1 \left(\frac{1}{r} + \frac{R}{r^2}\right) \]

### Step 7: Substitute \( Q_1 \)
Now substituting \( Q_1 = \frac{Qr^2}{r^2 + R^2} \):
\[ V_{\text{total}} = K \left(\frac{Qr^2}{r^2 + R^2}\right) \left(\frac{1}{r} + \frac{R}{r^2}\right) \]
Simplifying this gives us the final expression for the potential at the center.

### Final Expression
The potential at the common center can be expressed as:
\[ V = \frac{KQ}{r^2 + R^2} \left( R + r \right) \]