An isolated solid metal sphere of radius `R` is given an electric charge. The variation of the intensity of the electric field with the distance `r` from the centre of the sphere is best shown by

Options

A.
B.
C.
D.

Answer

To solve the problem regarding the variation of the electric field intensity with distance from the center of an isolated solid metal sphere of radius \( R \) that is given an electric charge, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Sphere's Properties**:
- An isolated solid metal sphere, when given a charge \( Q \), will have its charge distributed uniformly on its surface.

2. **Electric Field Inside the Sphere**:
- For any point inside the sphere (where \( r < R \)), the electric field intensity \( E \) is zero. This is because the electric field inside a conductor in electrostatic equilibrium is always zero.

\[ E = 0 \quad \text{for} \quad r < R \]

3. **Electric Field on the Surface of the Sphere**:
- At the surface of the sphere (where \( r = R \)), we can calculate the electric field using Gauss's Law. The electric field intensity \( E \) at the surface is given by:

\[ E = \frac{1}{4\pi \epsilon_0} \frac{Q}{R^2} \]

This value is the maximum electric field intensity for the sphere.

4. **Electric Field Outside the Sphere**:
- For points outside the sphere (where \( r > R \)), the electric field intensity behaves as if all the charge were concentrated at a point at the center of the sphere. The formula for the electric field intensity at a distance \( r \) from the center is:

\[ E = \frac{1}{4\pi \epsilon_0} \frac{Q}{r^2} \]

This shows that the electric field intensity decreases with the square of the distance from the center of the sphere.

5. **Graphical Representation**:
- The graph of electric field intensity \( E \) versus distance \( r \) will show:
- \( E = 0 \) for \( r < R \)
- A maximum value at \( r = R \)
- A decreasing curve for \( r > R \)

6. **Conclusion**:
- Therefore, the correct representation of the variation of electric field intensity with distance from the center of the sphere is best shown by a graph that starts at zero, reaches a maximum at the surface, and then decreases as you move further away.

### Final Answer:
The correct option for the variation of the electric field intensity with distance \( r \) from the center of the sphere is represented by option 3 (as mentioned in the video).