Two concentric coducting thin spherical shells A and B having radii `r_A` and `r_B(r_Bgtr_A)` are charged to `Q_A` and `-Q_B(|Q_B|gt|Q_A|)`. The electrical field along a line passing through the centre is

Options

A.
B.
C.
D. none of these

Explanation

To solve the problem of finding the electric field along a line passing through the center of two concentric conducting thin spherical shells A and B, we can break it down into several steps:

### Step 1: Understand the Configuration
We have two concentric conducting spherical shells:
- Shell A with radius \( r_A \) and charge \( Q_A \)
- Shell B with radius \( r_B \) (where \( r_B > r_A \)) and charge \( -Q_B \) (with \( |Q_B| > |Q_A| \))

### Step 2: Electric Field Inside Shell A
According to electrostatic principles, the electric field inside a conductor (and thus inside shell A) is zero. Therefore, for any point inside shell A (i.e., for \( r < r_A \)), the electric field \( E \) is:
\[ E = 0 \quad \text{for } r < r_A \]

### Step 3: Electric Field Between Shells A and B
For points between the two shells (i.e., for \( r_A < r < r_B \)), we can use Gauss's Law. The charge enclosed by a Gaussian surface in this region is only the charge on shell A, \( Q_A \). Thus, the electric field \( E \) can be calculated as:
\[ E \cdot 4\pi r^2 = \frac{Q_A}{\epsilon_0} \]
Solving for \( E \):
\[ E = \frac{Q_A}{4\pi \epsilon_0 r^2} \quad \text{for } r_A < r < r_B \]

### Step 4: Electric Field Outside Shell B
For points outside shell B (i.e., for \( r > r_B \)), we consider the total charge enclosed by a Gaussian surface that encompasses both shells. The total charge is:
\[ Q_{\text{total}} = Q_A - Q_B \]
Since \( |Q_B| > |Q_A| \), this results in a net negative charge. Thus, the electric field \( E \) is given by:
\[ E \cdot 4\pi r^2 = \frac{Q_A - Q_B}{\epsilon_0} \]
Solving for \( E \):
\[ E = \frac{Q_A - Q_B}{4\pi \epsilon_0 r^2} \quad \text{for } r > r_B \]

### Summary of Electric Field in Different Regions
- For \( r < r_A \): \( E = 0 \)
- For \( r_A < r < r_B \): \( E = \frac{Q_A}{4\pi \epsilon_0 r^2} \)
- For \( r > r_B \): \( E = \frac{Q_A - Q_B}{4\pi \epsilon_0 r^2} \)