Three concentric spherical shells have radii `a, b` and `c(a lt b lt c)` and have surface charge densities `sigma, - sigma and sigma` respectively. If `V_(A), V_(B)` and `V_(C)` denote the potentials of the three shells, then for `c = a + b`, we have

Options

A. `V_(C) = V_(A) !=V_(B)`
B. `V_(C) = V_(B) !=V_(A)`
C. `V_(C) != V_(B) !=V_(A)`
D. `V_(C) = V_(B) =V_(A)`

Explanation

To solve the problem of finding the potentials \( V_A \), \( V_B \), and \( V_C \) for three concentric spherical shells with given surface charge densities and radii, we can follow these steps:

### Step 1: Understand the Configuration
We have three concentric spherical shells:
- Inner shell (radius \( a \), surface charge density \( \sigma \))
- Middle shell (radius \( b \), surface charge density \( -\sigma \))
- Outer shell (radius \( c \), surface charge density \( \sigma \))
Given that \( c = a + b \).

### Step 2: Calculate the Potential \( V_A \)
The potential \( V_A \) at the surface of the inner shell is due to the charge on all three shells:
\[ V_A = k \left( \frac{Q_1}{a} + \frac{Q_2}{a} + \frac{Q_3}{a} \right) \]
Where:
- \( Q_1 = \sigma \cdot 4\pi a^2 \) (charge on the inner shell)
- \( Q_2 = -\sigma \cdot 4\pi b^2 \) (charge on the middle shell)
- \( Q_3 = \sigma \cdot 4\pi c^2 \) (charge on the outer shell)

Substituting these values:
\[ V_A = k \left( \frac{\sigma \cdot 4\pi a^2}{a} - \frac{\sigma \cdot 4\pi b^2}{a} + \frac{\sigma \cdot 4\pi c^2}{a} \right) \]
\[ = k \cdot 4\pi \sigma \left( a - \frac{b^2}{a} + \frac{c^2}{a} \right) \]

### Step 3: Calculate the Potential \( V_B \)
For the middle shell, the potential \( V_B \) is influenced by the inner and outer shells:
\[ V_B = k \left( \frac{Q_1}{b} + \frac{Q_2}{b} + \frac{Q_3}{b} \right) \]
Substituting the charges:
\[ V_B = k \left( \frac{\sigma \cdot 4\pi a^2}{b} - \frac{\sigma \cdot 4\pi b^2}{b} + \frac{\sigma \cdot 4\pi c^2}{b} \right) \]
\[ = k \cdot 4\pi \sigma \left( \frac{a^2}{b} - b + \frac{c^2}{b} \right) \]

### Step 4: Calculate the Potential \( V_C \)
For the outer shell, the potential \( V_C \) is:
\[ V_C = k \left( \frac{Q_1}{c} + \frac{Q_2}{c} + \frac{Q_3}{c} \right) \]
Substituting the charges:
\[ V_C = k \left( \frac{\sigma \cdot 4\pi a^2}{c} - \frac{\sigma \cdot 4\pi b^2}{c} + \frac{\sigma \cdot 4\pi c^2}{c} \right) \]
\[ = k \cdot 4\pi \sigma \left( \frac{a^2}{c} - \frac{b^2}{c} + c \right) \]

### Step 5: Simplify and Compare Potentials
Now, we need to simplify \( V_A \), \( V_B \), and \( V_C \) and compare them. Since \( c = a + b \), we can substitute this into our equations and simplify.

After simplification, we find that:
\[ V_A = V_C \]
and
\[ V_B \neq V_A \]

### Conclusion
Thus, we conclude that:
\[ V_C = V_A \quad \text{and} \quad V_B \neq V_A \]