Three concentric metallic spherical shells `A,B` and `C` of radii `a,b` and `c(a lt blt c)` have surface charge densities `+sigma ,-sigma `and `+sigma` respectively ,the potential of shell `B` is

Options

A. `sigma/epsi_(0) [(b^(2)-c^(2))/b+a]`
B. `sigma/epsi_(0) [(b^(2)-c^(2))/c+a]`
C. `sigma/epsi_(0) [(a^(2)-b^(2))/a+c]`
D. `sigma/epsi_(0) [(a^(2)-b^(2))/b+c]`

Answer

To find the potential at shell B due to the three concentric metallic spherical shells A, B, and C, we can follow these steps:

### Step 1: Understand the Configuration
We have three concentric metallic spherical shells:
- Shell A with radius \( a \) and surface charge density \( +\sigma \)
- Shell B with radius \( b \) and surface charge density \( -\sigma \)
- Shell C with radius \( c \) and surface charge density \( +\sigma \)

Given the relation \( a < b < c \).

### Step 2: Determine the Potential at Shell B
The potential at a point due to a charged spherical shell can be calculated using the following principles:
1. The potential inside a conducting shell is constant and equal to the potential on its surface.
2. The potential outside a charged shell behaves as if all the charge were concentrated at the center.

### Step 3: Calculate the Contribution to Potential at Shell B
The potential \( V_B \) at shell B will be the sum of the potentials due to shells A, B, and C.

1. **Potential due to Shell A** (outside point for A):
\[ V_A = \frac{k Q_A}{b} \]
where \( Q_A = \sigma \cdot 4\pi a^2 \).

2. **Potential due to Shell B** (on the surface):
\[ V_B = \frac{k Q_B}{b} \]
where \( Q_B = -\sigma \cdot 4\pi b^2 \).

3. **Potential due to Shell C** (inside point for C):
\[ V_C = \frac{k Q_C}{c} \]
where \( Q_C = \sigma \cdot 4\pi c^2 \).

### Step 4: Substitute the Charge Values
Now substituting the charge values into the potential equations:
1. For shell A:
\[ V_A = \frac{k (\sigma \cdot 4\pi a^2)}{b} \]

2. For shell B:
\[ V_B = \frac{k (-\sigma \cdot 4\pi b^2)}{b} = -k \sigma \cdot 4\pi b \]

3. For shell C:
\[ V_C = \frac{k (\sigma \cdot 4\pi c^2)}{c} = k \sigma \cdot 4\pi c \]

### Step 5: Combine the Potentials
Now, we can combine all the contributions:
\[ V_B = V_A + V_B + V_C \]
Substituting the values:
\[ V_B = \frac{k \sigma \cdot 4\pi a^2}{b} - k \sigma \cdot 4\pi b + k \sigma \cdot 4\pi c \]

### Step 6: Factor Out Common Terms
Factoring out \( k \sigma \cdot 4\pi \):
\[ V_B = k \sigma \cdot 4\pi \left( \frac{a^2}{b} - b + c \right) \]

### Step 7: Final Expression
Thus, the potential at shell B is given by:
\[ V_B = \frac{4\pi k \sigma}{\epsilon_0} \left( \frac{a^2}{b} - b + c \right) \]