Three concentric metallic spherical shell A,B and C or radii a,b and c`(a lt b lt c)` have surface charge densities `-sigma, + sigma and -sigma`. Respectively. The potential of shel A is :

Options

A. `(sigma//epsilon_(0))(a+b-c)`
B. `(sigma//epsilon_(0))(a-b+c)`
C. `(sigma//epsilon_(0))(a-b-c)`
D. none

Answer

To find the potential at shell A, we will follow these steps:

### Step 1: Understanding the Configuration
We have three concentric metallic spherical shells A, B, and C with radii \( a \), \( b \), and \( c \) respectively, where \( a < b < c \). The surface charge densities are given as follows:
- Shell A: \( -\sigma \)
- Shell B: \( +\sigma \)
- Shell C: \( -\sigma \)

### Step 2: Formula for Potential
The potential \( V \) at a point due to a charged shell can be calculated using the formula:
\[ V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r} \]
where \( Q \) is the charge on the shell and \( r \) is the distance from the center to the point where the potential is being calculated.

### Step 3: Calculate Charges on the Shells
The charge \( Q \) on each shell can be calculated using the formula:
\[ Q = \sigma \times A \]
where \( A \) is the surface area of the shell. The surface area of a sphere is given by \( 4\pi r^2 \).

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

### Step 4: Calculate Potential at Shell A
The potential at shell A due to itself, shell B, and shell C can be expressed as:
\[ V_A = V_{A} + V_{B} + V_{C} \]
Since the potential due to a shell at its own surface is constant and equal to the potential outside it, we have:
\[ V_A = \frac{1}{4\pi\epsilon_0} \left( Q_A \cdot \frac{1}{a} + Q_B \cdot \frac{1}{a} + Q_C \cdot \frac{1}{a} \right) \]
Substituting the charges:
\[ V_A = \frac{1}{4\pi\epsilon_0} \left( -4\pi\sigma a^2 \cdot \frac{1}{a} + 4\pi\sigma b^2 \cdot \frac{1}{a} - 4\pi\sigma c^2 \cdot \frac{1}{a} \right) \]

### Step 5: Simplifying the Expression
Now, simplifying the expression:
\[ V_A = \frac{1}{4\pi\epsilon_0} \cdot \frac{-4\pi\sigma a^2 + 4\pi\sigma b^2 - 4\pi\sigma c^2}{a} \]
\[ = \frac{-\sigma a + \sigma b - \sigma c}{\epsilon_0} \]
\[ = \frac{\sigma}{\epsilon_0} \left( -a + b - c \right) \]

### Final Answer
Thus, the potential at shell A is:
\[ V_A = \frac{\sigma}{\epsilon_0} (b - a - c) \]