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 = q + b`, we have

Options

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

Answer

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, we can follow these steps:

### Step 1: Understand the Setup
We have three concentric spherical shells with radii \( a \), \( b \), and \( c \) (where \( a < b < c \)). The surface charge densities are:
- Shell at radius \( a \): \( \sigma \)
- Shell at radius \( b \): \( -\sigma \)
- Shell at radius \( c \): \( \sigma \)

### Step 2: Calculate the Potential \( V_A \)
The potential \( V_A \) at radius \( a \) is due to the charges on all three shells. The contributions to the potential from each shell are:
- From shell at \( a \): \( V_a = k \cdot \sigma \cdot 4\pi a^2 / a = k \cdot \sigma \cdot 4\pi a \)
- From shell at \( b \): \( V_b = k \cdot (-\sigma) \cdot 4\pi b^2 / a = -k \cdot \sigma \cdot 4\pi b^2 / a \)
- From shell at \( c \): \( V_c = k \cdot \sigma \cdot 4\pi c^2 / a = k \cdot \sigma \cdot 4\pi c^2 / a \)

Thus, the total potential at point \( A \) is:
\[ V_A = k \cdot \sigma \cdot 4\pi a - k \cdot \sigma \cdot \frac{4\pi b^2}{a} + k \cdot \sigma \cdot \frac{4\pi c^2}{a} \]
Factoring out \( k \cdot \sigma \cdot 4\pi \):
\[ V_A = k \cdot \sigma \cdot 4\pi \left( a - \frac{b^2}{a} + \frac{c^2}{a} \right) \]

### Step 3: Calculate the Potential \( V_B \)
For the potential \( V_B \) at radius \( b \), we consider contributions from the shells at \( a \) and \( c \):
- From shell at \( a \): \( V_a = k \cdot \sigma \cdot 4\pi a \)
- From shell at \( b \): \( V_b = -\sigma \cdot 4\pi b^2 / b = -k \cdot \sigma \cdot 4\pi b \)
- From shell at \( c \): \( V_c = k \cdot \sigma \cdot 4\pi c^2 / b \)

Thus, the total potential at point \( B \) is:
\[ V_B = k \cdot \sigma \cdot 4\pi a - k \cdot \sigma \cdot 4\pi b + k \cdot \sigma \cdot \frac{4\pi c^2}{b} \]
Factoring out \( k \cdot \sigma \cdot 4\pi \):
\[ V_B = k \cdot \sigma \cdot 4\pi \left( a - b + \frac{c^2}{b} \right) \]

### Step 4: Calculate the Potential \( V_C \)
For the potential \( V_C \) at radius \( c \), we consider contributions from all shells:
- From shell at \( a \): \( V_a = k \cdot \sigma \cdot 4\pi a \)
- From shell at \( b \): \( V_b = -k \cdot \sigma \cdot 4\pi b \)
- From shell at \( c \): \( V_c = k \cdot \sigma \cdot 4\pi c \)

Thus, the total potential at point \( C \) is:
\[ V_C = k \cdot \sigma \cdot 4\pi a - k \cdot \sigma \cdot 4\pi b + k \cdot \sigma \cdot 4\pi c \]
Factoring out \( k \cdot \sigma \cdot 4\pi \):
\[ V_C = k \cdot \sigma \cdot 4\pi \left( a - b + c \right) \]

### Step 5: Analyze the Relationship
Given that \( c = a + b \), we can substitute \( c \) into our expressions for \( V_A \), \( V_B \), and \( V_C \):
- From the expression for \( V_A \):
\[ V_A = k \cdot \sigma \cdot 4\pi \left( a - \frac{b^2}{a} + \frac{(a+b)^2}{a} \right) \]
- From the expression for \( V_B \):
\[ V_B = k \cdot \sigma \cdot 4\pi \left( a - b + \frac{(a+b)^2}{b} \right) \]
- From the expression for \( V_C \):
\[ V_C = k \cdot \sigma \cdot 4\pi \left( a - b + (a+b) \right) = k \cdot \sigma \cdot 4\pi \left( 2a \right) \]

### Conclusion
From the calculations, we find that:
- \( V_A \) and \( V_C \) are equal.
- \( V_B \) is different from \( V_A \) and \( V_C \).