A conducting sphere of radius 10 cm has unknown charge. If the electric field at a distance 20 cm from the centre of the sphere is `1.2xx10^(3)NC^(-1)` and points radially inwards. The net charge on the sphere is

Options

A. `-4.5xx10^(-9)C`
B. `4.5xx10^(9)C`
C. `-5.3xx10^(-9)C`
D. `5.3xx10^(9)C`

Explanation

To find the net charge on a conducting sphere given the electric field at a certain distance from its center, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of the sphere, \( r = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Distance from the center where the electric field is measured, \( d = 20 \, \text{cm} = 0.2 \, \text{m} \)
- Electric field, \( E = 1.2 \times 10^3 \, \text{N/C} \)

2. **Determine the Direction of the Electric Field:**
- The electric field points radially inwards, indicating that the charge on the sphere is negative.

3. **Use the Formula for Electric Field due to a Point Charge:**
The electric field \( E \) at a distance \( d \) from a point charge \( Q \) is given by:
\[ E = \frac{1}{4\pi \epsilon_0} \frac{Q}{d^2} \]
where \( \epsilon_0 \) (the permittivity of free space) is approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \).

4. **Rearranging the Formula to Solve for Charge \( Q \):**
\[ Q = E \cdot 4\pi \epsilon_0 \cdot d^2 \]

5. **Substituting the Values:**
- Calculate \( 4\pi \epsilon_0 \):
\[ 4\pi \epsilon_0 \approx 4 \times 3.14 \times 8.85 \times 10^{-12} \approx 1.11 \times 10^{-10} \, \text{C}^2/\text{N m}^2 \]
- Substitute \( E \) and \( d \):
\[ Q = (1.2 \times 10^3) \cdot (1.11 \times 10^{-10}) \cdot (0.2^2) \]
- Calculate \( d^2 \):
\[ d^2 = (0.2)^2 = 0.04 \, \text{m}^2 \]
- Now substituting back:
\[ Q = (1.2 \times 10^3) \cdot (1.11 \times 10^{-10}) \cdot (0.04) \]

6. **Perform the Calculation:**
\[ Q = (1.2 \times 10^3) \cdot (1.11 \times 10^{-10}) \cdot (0.04) = 5.33 \times 10^{-9} \, \text{C} \]

7. **Indicate the Sign of the Charge:**
Since the electric field is directed inwards, the charge is negative:
\[ Q = -5.33 \times 10^{-9} \, \text{C} \]

### Final Answer:
The net charge on the sphere is:
\[ Q = -5.33 \times 10^{-9} \, \text{C} \]