The number of electrons to be put on a spherical conductor of radius 0.1m to produce an electric field of 0.036N/C just above its surface is

Options

A. `2.7 xx 10^(5)`
B. `2.6 xx 10^(5)`
C. `2.5 xx 10^(5)`
D. `2.4 xx 10^(5)`

Answer

To solve the problem of determining the number of electrons needed to produce an electric field of 0.036 N/C just above the surface of a spherical conductor with a radius of 0.1 m, we can follow these steps:

### Step 1: Understand the relationship between electric field and charge
The electric field \( E \) just outside the surface of a spherical conductor is given by the formula:
\[ E = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2} \]
where:
- \( E \) is the electric field,
- \( Q \) is the total charge on the surface of the conductor,
- \( r \) is the radius of the sphere,
- \( \epsilon_0 \) is the permittivity of free space (\( \epsilon_0 \approx 8.854 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)).

### Step 2: Express total charge \( Q \) in terms of the number of electrons
Let \( m \) be the number of electrons. The total charge \( Q \) can be expressed as:
\[ Q = m \cdot e \]
where \( e \) is the charge of an electron (\( e \approx 1.6 \times 10^{-19} \, \text{C} \)).

### Step 3: Substitute \( Q \) into the electric field equation
Substituting \( Q \) into the electric field equation gives:
\[ E = \frac{1}{4 \pi \epsilon_0} \frac{m \cdot e}{r^2} \]

### Step 4: Rearrange the equation to solve for \( m \)
Rearranging the equation to solve for \( m \):
\[ m = \frac{E \cdot 4 \pi \epsilon_0 \cdot r^2}{e} \]

### Step 5: Plug in the known values
Substituting the known values:
- \( E = 0.036 \, \text{N/C} \)
- \( r = 0.1 \, \text{m} \)
- \( \epsilon_0 \approx 8.854 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)
- \( e \approx 1.6 \times 10^{-19} \, \text{C} \)

Calculating \( m \):
\[ m = \frac{0.036 \cdot 4 \pi (8.854 \times 10^{-12}) \cdot (0.1)^2}{1.6 \times 10^{-19}} \]

### Step 6: Calculate the numerical value
Calculating the numerator:
\[ 4 \pi (8.854 \times 10^{-12}) \cdot (0.1)^2 \approx 1.112 \times 10^{-12} \]
Thus,
\[ m = \frac{0.036 \cdot 1.112 \times 10^{-12}}{1.6 \times 10^{-19}} \approx \frac{4.0032 \times 10^{-14}}{1.6 \times 10^{-19}} \approx 2.502 \times 10^{5} \]

### Step 7: Round off the answer
Rounding off gives:
\[ m \approx 2.5 \times 10^{5} \]

### Final Answer
The number of electrons to be put on the spherical conductor is approximately \( 2.5 \times 10^{5} \).

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