The correctness of velocity of an electron moving with velocity `50 ms^(-1)` is `0.005%` . The accuracy with which its position can be measured will be

Options

A. `4634xx10^(-3)m`
B. `4634xx10^(-5)m`
C. `4634xx10^(-6)m`
D. `4634xx10^(-8)m`

Answer

To solve the problem of determining the accuracy with which the position of an electron can be measured given its velocity and the correctness of that velocity, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Velocity of the electron, \( V = 50 \, \text{ms}^{-1} \)
- Correctness of velocity, \( \Delta V / V \times 100 = 0.005\% \)

2. **Convert the Correctness Percentage to Decimal**:
- Convert \( 0.005\% \) to decimal form:
\[ 0.005\% = \frac{0.005}{100} = 0.00005 \]

3. **Calculate the Uncertainty in Velocity (\( \Delta V \))**:
- Use the formula for correctness:
\[ \Delta V = V \times 0.00005 \]
- Substitute the value of \( V \):
\[ \Delta V = 50 \times 0.00005 = 0.0025 \, \text{ms}^{-1} \]

4. **Use the Heisenberg Uncertainty Principle**:
- The uncertainty in position (\( \Delta X \)) is related to the uncertainty in momentum (\( \Delta P \)) by the formula:
\[ \Delta X \Delta P \geq \frac{h}{4\pi} \]
- Where \( \Delta P = m \Delta V \) and \( h \) is Planck's constant (\( 6.63 \times 10^{-34} \, \text{Js} \)).
- The mass of the electron \( m \) is \( 9.1 \times 10^{-31} \, \text{kg} \).

5. **Calculate the Uncertainty in Momentum (\( \Delta P \))**:
- Calculate \( \Delta P \):
\[ \Delta P = m \Delta V = (9.1 \times 10^{-31}) \times (0.0025) = 2.275 \times 10^{-33} \, \text{kg m/s} \]

6. **Calculate the Uncertainty in Position (\( \Delta X \))**:
- Rearranging the Heisenberg uncertainty principle gives:
\[ \Delta X \geq \frac{h}{4\pi \Delta P} \]
- Substitute the values:
\[ \Delta X \geq \frac{6.63 \times 10^{-34}}{4\pi (2.275 \times 10^{-33})} \]
- Calculate \( \Delta X \):
\[ \Delta X \geq \frac{6.63 \times 10^{-34}}{4 \times 3.14 \times 2.275 \times 10^{-33}} \approx 4.63 \times 10^{-5} \, \text{m} \]

### Final Result:
The accuracy with which the position of the electron can be measured is approximately \( 4.63 \times 10^{-5} \, \text{m} \) or \( 4634 \times 10^{-5} \, \text{m} \).