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. `2.7xx10^(5)`
B. `2.6xx10^(5)`
C. `2.5xx10^(5)`
D. `2.4xx10^(5)`

Explanation

To find the number of electrons that need to be put on a spherical conductor of radius \(0.1 \, \text{m}\) to produce an electric field of \(0.036 \, \text{N/C}\) just above its surface, 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 charged spherical conductor is given by the formula:
\[ E = \frac{k \cdot Q}{r^2} \]
where:
- \(E\) is the electric field,
- \(k\) is Coulomb's constant (\(9 \times 10^9 \, \text{N m}^2/\text{C}^2\)),
- \(Q\) is the total charge on the conductor,
- \(r\) is the radius of the conductor.

### Step 2: Rearranging the formula to find \(Q\)
We can rearrange the formula to solve for \(Q\):
\[ Q = \frac{E \cdot r^2}{k} \]

### Step 3: Substitute the known values
Given:
- \(E = 0.036 \, \text{N/C}\)
- \(r = 0.1 \, \text{m}\)

Substituting these values into the equation:
\[ Q = \frac{0.036 \cdot (0.1)^2}{9 \times 10^9} \]

### Step 4: Calculate \(Q\)
Calculating \(Q\):
\[ Q = \frac{0.036 \cdot 0.01}{9 \times 10^9} = \frac{0.00036}{9 \times 10^9} = 4 \times 10^{-11} \, \text{C} \]

### Step 5: Calculate the number of electrons
The charge of a single electron is approximately \(1.6 \times 10^{-19} \, \text{C}\). To find the number of electrons \(n\), we use the formula:
\[ n = \frac{Q}{e} \]
where \(e\) is the charge of one electron.

Substituting the values:
\[ n = \frac{4 \times 10^{-11}}{1.6 \times 10^{-19}} \approx 2.5 \times 10^8 \]

### Final Answer
The number of electrons to be put on the spherical conductor is approximately \(2.5 \times 10^8\).

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