Three concentric spherical metallic shells A, B and C of radii a, b and c (a < b < c) have surface charge densities `sigma`, `-sigma` and `sigma` respectively.
`(i) Find the potential of the three shells A, B and C.
(ii) If the shells A and C are at the same potential, obtain the relation between the radii a, b and c.

Explanation

To solve the given problem, we will find the potentials of the three concentric spherical metallic shells A, B, and C, and then derive the relationship between their radii if the potentials of shells A and C are equal.

### Step 1: Calculate the Charge on Each Shell
The surface charge densities are given as follows:
- Shell A has surface charge density \( \sigma \)
- Shell B has surface charge density \( -\sigma \)
- Shell C has surface charge density \( \sigma \)

The charges on the shells can be calculated using the formula:
\[ Q = \sigma \times \text{Surface Area} \]
The surface area of a sphere is given by \( 4\pi r^2 \).

Thus, the charges are:
- Charge on shell A:
\[ Q_A = \sigma \times 4\pi a^2 \]
- Charge on shell B:
\[ Q_B = -\sigma \times 4\pi b^2 \]
- Charge on shell C:
\[ Q_C = \sigma \times 4\pi c^2 \]

### Step 2: Calculate the Potential of Shell A
The potential \( V_A \) at a point inside shell A (which is also inside shells B and C) is given by:
\[ V_A = K \cdot Q_A \cdot \frac{1}{a} + V_B + V_C \]
Where \( K = \frac{1}{4\pi \epsilon_0} \).

Since the potential due to a shell is constant inside it, we can express the potential as:
\[ V_A = K \cdot Q_A \cdot \frac{1}{a} + K \cdot Q_B \cdot \frac{1}{b} + K \cdot Q_C \cdot \frac{1}{c} \]
Substituting the values of \( Q_A \), \( Q_B \), and \( Q_C \):
\[ V_A = \frac{\sigma}{\epsilon_0} \left( a + b + c \right) \]

### Step 3: Calculate the Potential of Shell B
For shell B, which is between shells A and C, the potential \( V_B \) is given by:
\[ V_B = K \cdot Q_A \cdot \frac{1}{b} + K \cdot Q_C \cdot \frac{1}{c} \]
Substituting the values:
\[ V_B = \frac{\sigma}{\epsilon_0} \left( \frac{a^2}{b} + \frac{c^2}{c} \right) \]

### Step 4: Calculate the Potential of Shell C
For shell C, which is outside both shells A and B, the potential \( V_C \) is given by:
\[ V_C = K \cdot (Q_A + Q_B + Q_C) \cdot \frac{1}{c} \]
Substituting the values:
\[ V_C = \frac{\sigma}{\epsilon_0} \left( \frac{a^2 + b^2 + c^2}{c} \right) \]

### Step 5: Equate Potentials of Shell A and Shell C
If the potentials of shells A and C are equal, we have:
\[ V_A = V_C \]
Substituting the expressions for \( V_A \) and \( V_C \):
\[ \frac{\sigma}{\epsilon_0} (a + b + c) = \frac{\sigma}{\epsilon_0} \left( \frac{a^2 + b^2 + c^2}{c} \right) \]
Cancelling \( \frac{\sigma}{\epsilon_0} \) from both sides:
\[ a + b + c = \frac{a^2 + b^2 + c^2}{c} \]
Multiplying through by \( c \):
\[ c(a + b + c) = a^2 + b^2 + c^2 \]
Rearranging gives:
\[ a^2 + b^2 + c^2 - ac - bc - c^2 = 0 \]
This simplifies to:
\[ c = a + b \]

### Final Answers
(i) The potentials of the shells are:
- \( V_A = \frac{\sigma}{\epsilon_0} (a + b + c) \)
- \( V_B = \frac{\sigma}{\epsilon_0} \left( \frac{a^2}{b} + c \right) \)
- \( V_C = \frac{\sigma}{\epsilon_0} \left( \frac{a^2 + b^2 + c^2}{c} \right) \)

(ii) The relation between the radii is:
\[ c = a + b \]