At a point `20 cm` from the centre of a uniformly charged dielectric sphere of radius `10 cm`, the electric field is `100 V//m`. The electric field at `3 cm` from the centre of the sphere will be

Options

A. `150 V//m`
B. `125V//m`
C. `120V//m`
D. Zero

Explanation

To solve the problem, we need to find the electric field at a distance of 3 cm from the center of a uniformly charged dielectric sphere, given that the electric field at a distance of 20 cm from the center is 100 V/m.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a uniformly charged dielectric sphere with a radius of 10 cm. We know the electric field at a point 20 cm from the center of the sphere is 100 V/m. We need to find the electric field at a point 3 cm from the center of the sphere.

2. **Identify the Regions**:
- The point at 20 cm is outside the sphere (since 20 cm > 10 cm).
- The point at 3 cm is inside the sphere (since 3 cm < 10 cm).

3. **Electric Field Outside the Sphere**:
For points outside the sphere (r > R), the electric field \( E \) is given by:
\[ E = \frac{kQ}{r^2} \]
where \( k \) is Coulomb's constant, \( Q \) is the total charge of the sphere, and \( r \) is the distance from the center.

From the problem, we know:
\[ E_{20} = 100 \, \text{V/m} \quad \text{at} \quad r = 20 \, \text{cm} = 0.2 \, \text{m} \]

4. **Calculate the Charge \( Q \)**:
Using the electric field formula at 20 cm:
\[ 100 = \frac{kQ}{(0.2)^2} \]
Rearranging gives:
\[ kQ = 100 \times (0.2)^2 = 100 \times 0.04 = 4 \, \text{N m}^2/\text{C} \]

5. **Electric Field Inside the Sphere**:
For points inside the sphere (r < R), the electric field \( E \) is given by:
\[ E = \frac{kQx}{R^3} \]
where \( x \) is the distance from the center and \( R \) is the radius of the sphere.

6. **Substituting Values**:
We need to find the electric field at \( x = 3 \, \text{cm} = 0.03 \, \text{m} \) and \( R = 10 \, \text{cm} = 0.1 \, \text{m} \):
\[ E_{3} = \frac{kQ \cdot 0.03}{(0.1)^3} \]
Substituting \( kQ = 4 \):
\[ E_{3} = \frac{4 \cdot 0.03}{(0.1)^3} = \frac{4 \cdot 0.03}{0.001} = 4 \cdot 30 = 120 \, \text{V/m} \]

7. **Final Answer**:
The electric field at a distance of 3 cm from the center of the sphere is **120 V/m**.