A thin spherical shell of radius R has charge Q spread uniformly over its surface. Which of the following graphs most closely represents the electric field E(r) produced by the shell in the range `0lerltoo`, where r is the distance from the centre of the shell?

Options

A.
B.
C.
D.

Answer

To solve the problem of determining the electric field \( E(r) \) produced by a thin spherical shell of radius \( R \) with charge \( Q \) uniformly spread over its surface, we can analyze the electric field in different regions relative to the shell.

### Step-by-Step Solution:

1. **Understanding the Shell's Charge Distribution**:
- The charge \( Q \) is uniformly distributed over the surface of a thin spherical shell of radius \( R \).
- Inside the shell (for \( r < R \)), there are no charges present.

2. **Applying Gauss's Law Inside the Shell**:
- According to Gauss's Law, the electric field \( E \) inside a uniformly charged shell is zero.
- Therefore, for \( 0 < r < R \):
\[ E(r) = 0 \]

3. **Analyzing the Electric Field Outside the Shell**:
- For \( r > R \), we can consider a Gaussian surface in the form of a sphere of radius \( r \) centered at the center of the shell.
- The total charge enclosed by this Gaussian surface is \( Q \).
- By Gauss's Law:
\[ \oint \vec{E} \cdot d\vec{A} = \frac{Q}{\epsilon_0} \]
- Since the electric field \( E \) is uniform over the surface of the sphere, we can write:
\[ E \cdot 4\pi r^2 = \frac{Q}{\epsilon_0} \]
- Solving for \( E \):
\[ E(r) = \frac{Q}{4\pi \epsilon_0 r^2} \]

4. **Graphing the Electric Field**:
- From the analysis, we find that:
- For \( 0 < r < R \): \( E(r) = 0 \)
- For \( r \geq R \): \( E(r) \) decreases with \( 1/r^2 \).
- This means the electric field is zero inside the shell and decreases as we move away from the shell.

5. **Identifying the Correct Graph**:
- The graph that represents this behavior will show:
- A flat line at \( E = 0 \) for \( 0 < r < R \).
- A curve that decreases as \( r \) increases for \( r \geq R \).

### Conclusion:
The graph that most closely represents the electric field \( E(r) \) produced by the shell is the one that shows \( E = 0 \) for \( r < R \) and a decreasing curve for \( r \geq R \).