Three concentric spherical metallic spheres `A,B` and `C` of radii `a , b` and `c(a lt b lt c)` have surface charge densities `sigma , -sigma` and `sigma` respectively.

Options

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

Answer

To solve the problem involving three concentric spherical metallic spheres A, B, and C with given surface charge densities, we will calculate the electric potential at the surfaces of each sphere. The spheres have radii \( a \), \( b \), and \( c \) respectively, with \( a < b < c \) and surface charge densities \( \sigma \), \( -\sigma \), and \( \sigma \).

### Step-by-Step Solution:

1. **Identify Charge on Each Sphere:**
- The charge on sphere A, \( Q_A \), is given by:
\[ Q_A = \sigma \cdot 4\pi a^2 \]
- The charge on sphere B, \( Q_B \), is given by:
\[ Q_B = -\sigma \cdot 4\pi b^2 \]
- The charge on sphere C, \( Q_C \), is given by:
\[ Q_C = \sigma \cdot 4\pi c^2 \]

2. **Calculate Potential at Sphere A (VA):**
- The potential at sphere A due to its own charge and the charges of spheres B and C is:
\[ V_A = \frac{1}{4\pi \epsilon_0} \left( \frac{Q_A}{a} + \frac{Q_B}{b} + \frac{Q_C}{c} \right) \]
- Substituting the values of \( Q_A \), \( Q_B \), and \( Q_C \):
\[ V_A = \frac{1}{4\pi \epsilon_0} \left( \frac{\sigma \cdot 4\pi a^2}{a} + \frac{-\sigma \cdot 4\pi b^2}{b} + \frac{\sigma \cdot 4\pi c^2}{c} \right) \]
- Simplifying this expression:
\[ V_A = \frac{\sigma}{\epsilon_0} \left( a - b + c \right) \]

3. **Calculate Potential at Sphere B (VB):**
- The potential at sphere B is given by:
\[ V_B = \frac{1}{4\pi \epsilon_0} \left( \frac{Q_A}{b} + \frac{Q_B}{b} + \frac{Q_C}{c} \right) \]
- Substituting the values:
\[ V_B = \frac{1}{4\pi \epsilon_0} \left( \frac{\sigma \cdot 4\pi a^2}{b} + \frac{-\sigma \cdot 4\pi b^2}{b} + \frac{\sigma \cdot 4\pi c^2}{c} \right) \]
- Simplifying this expression:
\[ V_B = \frac{\sigma}{\epsilon_0} \left( \frac{a^2}{b} - b + c \right) \]

4. **Calculate Potential at Sphere C (VC):**
- The potential at sphere C is given by:
\[ V_C = \frac{1}{4\pi \epsilon_0} \left( \frac{Q_A}{c} + \frac{Q_B}{c} + \frac{Q_C}{c} \right) \]
- Substituting the values:
\[ V_C = \frac{1}{4\pi \epsilon_0} \left( \frac{\sigma \cdot 4\pi a^2}{c} + \frac{-\sigma \cdot 4\pi b^2}{c} + \frac{\sigma \cdot 4\pi c^2}{c} \right) \]
- Simplifying this expression:
\[ V_C = \frac{\sigma}{\epsilon_0} \left( \frac{a^2}{c} - \frac{b^2}{c} + c \right) \]

### Summary of Results:
- \( V_A = \frac{\sigma}{\epsilon_0} (a - b + c) \)
- \( V_B = \frac{\sigma}{\epsilon_0} \left( \frac{a^2}{b} - b + c \right) \)
- \( V_C = \frac{\sigma}{\epsilon_0} \left( \frac{a^2 - b^2}{c} + c \right) \)