Two electrons are fired towards each other with a speed of `3xx10^(6)` m/s. What would be the minimum separation between the electrons?

Options

A. `2.8 xx 10 ^(-11)`
B. `2.0 xx 10 ^(-11)`
C. `9.8 xx 10 ^(-11)`
D. NONE

Explanation

To find the minimum separation between two electrons fired towards each other with a speed of \(3 \times 10^6\) m/s, we can use the principle of conservation of energy. The initial kinetic energy of the electrons will be converted into electrostatic potential energy when they come to their closest approach.

### Step-by-Step Solution:

1. **Determine the Initial Kinetic Energy (KE):**
Each electron has a mass \(m\) and is moving with a speed \(v = 3 \times 10^6\) m/s. The kinetic energy of one electron is given by:
\[ KE = \frac{1}{2} mv^2 \]
Since there are two electrons, the total initial kinetic energy \(KE_{total}\) is:
\[ KE_{total} = 2 \times \frac{1}{2} mv^2 = mv^2 \]

2. **Determine the Electrostatic Potential Energy (PE) at Minimum Separation:**
The electrostatic potential energy between two point charges is given by:
\[ PE = \frac{k \cdot |q_1 \cdot q_2|}{r} \]
For two electrons, the charge \(q_1 = q_2 = e\), where \(e = 1.6 \times 10^{-19}\) C, and \(k\) is Coulomb's constant, approximately \(9 \times 10^9 \, \text{N m}^2/\text{C}^2\). Thus:
\[ PE = \frac{k \cdot e^2}{r} \]

3. **Apply Conservation of Energy:**
At the point of closest approach, all the kinetic energy will be converted into potential energy:
\[ KE_{total} = PE \]
Substituting the expressions for kinetic and potential energy, we have:
\[ mv^2 = \frac{k \cdot e^2}{r} \]

4. **Rearrange to Solve for Minimum Separation \(r\):**
Rearranging the equation gives:
\[ r = \frac{k \cdot e^2}{mv^2} \]

5. **Substitute the Known Values:**
Now substitute the known values:
- \(k = 9 \times 10^9 \, \text{N m}^2/\text{C}^2\)
- \(e = 1.6 \times 10^{-19} \, \text{C}\)
- \(m = 9.31 \times 10^{-31} \, \text{kg}\)
- \(v = 3 \times 10^6 \, \text{m/s}\)

Thus:
\[ r = \frac{(9 \times 10^9) \cdot (1.6 \times 10^{-19})^2}{(9.31 \times 10^{-31}) \cdot (3 \times 10^6)^2} \]

6. **Calculate the Value of \(r\):**
First, calculate the numerator:
\[ k \cdot e^2 = 9 \times 10^9 \cdot (1.6 \times 10^{-19})^2 = 9 \times 10^9 \cdot 2.56 \times 10^{-38} = 2.304 \times 10^{-28} \]
Then calculate the denominator:
\[ mv^2 = 9.31 \times 10^{-31} \cdot (3 \times 10^6)^2 = 9.31 \times 10^{-31} \cdot 9 \times 10^{12} = 8.379 \times 10^{-18} \]
Now, substituting these values into the equation for \(r\):
\[ r = \frac{2.304 \times 10^{-28}}{8.379 \times 10^{-18}} \approx 2.75 \times 10^{-11} \, \text{m} \]

### Final Result:
The minimum separation between the two electrons is approximately:
\[ r \approx 2.75 \times 10^{-11} \, \text{m} \]