`4xx10^(10)` electrons are removed from a neutral metal sphere of diameter 20 cm placed in air. The magnitude of the electric field (in `NC^(-1)`) at a distance of 20 cm from its centre is

Options

A. `460`
B. `5760`
C. Zero
D. `1440`

Answer

To solve the problem step by step, we will calculate the electric field at a distance of 20 cm from the center of a neutral metal sphere after removing a certain number of electrons.

### Step 1: Determine the charge on the sphere
When electrons are removed from a neutral metal sphere, the sphere becomes positively charged. The total charge \( Q \) can be calculated using the formula:

\[ Q = n \cdot e \]

where:
- \( n \) is the number of electrons removed, which is \( 4 \times 10^{10} \).
- \( e \) is the charge of a single electron, approximately \( 1.6 \times 10^{-19} \, \text{C} \).

Substituting the values:

\[ Q = 4 \times 10^{10} \times 1.6 \times 10^{-19} \, \text{C} \]

Calculating this gives:

\[ Q = 6.4 \times 10^{-9} \, \text{C} \]

### Step 2: Convert diameter to radius
The diameter of the sphere is given as 20 cm. To find the radius \( r \):

\[ r = \frac{d}{2} = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} = 0.1 \, \text{m} \]

### Step 3: Determine the distance from the center
The distance from the center to the point where we want to calculate the electric field is also given as 20 cm, which is:

\[ P = 20 \, \text{cm} = 0.2 \, \text{m} \]

### Step 4: Calculate the electric field
The electric field \( E \) at a distance \( P \) from the center of a charged sphere is given by the formula:

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

where:
- \( \epsilon_0 \) (the permittivity of free space) is approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \).

Substituting the values:

\[ E = \frac{6.4 \times 10^{-9}}{4 \pi (8.85 \times 10^{-12}) (0.2)^2} \]

Calculating \( 4 \pi \epsilon_0 \):

\[ 4 \pi \epsilon_0 \approx 4 \times 3.14 \times 8.85 \times 10^{-12} \approx 1.11 \times 10^{-10} \]

Now substituting this back into the electric field equation:

\[ E = \frac{6.4 \times 10^{-9}}{1.11 \times 10^{-10} \times 0.04} \]

Calculating the denominator:

\[ 1.11 \times 10^{-10} \times 0.04 \approx 4.44 \times 10^{-12} \]

Now calculating \( E \):

\[ E = \frac{6.4 \times 10^{-9}}{4.44 \times 10^{-12}} \approx 1440 \, \text{N/C} \]

### Final Answer
The magnitude of the electric field at a distance of 20 cm from the center of the sphere is approximately:

\[ \boxed{1440 \, \text{N/C}} \]