An electron has a speed of `40m//s`, accurate up `99.99%`.What is the uncertainty in locating position ?

Explanation

To solve the problem of finding the uncertainty in the position of an electron given its speed and accuracy, we can follow these steps:

### Step 1: Understand the Uncertainty Principle
According to Heisenberg's Uncertainty Principle, the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) is given by the formula:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where \( h \) is Planck's constant.

### Step 2: Calculate the Uncertainty in Velocity (Δv)
The speed of the electron is given as \( 40 \, \text{m/s} \) with an accuracy of \( 99.99\% \). To find the uncertainty in velocity (Δv), we calculate:
\[ \Delta v = \left(1 - \frac{99.99}{100}\right) \times 40 \, \text{m/s} \]
Calculating this gives:
\[ \Delta v = 0.0001 \times 40 = 0.004 \, \text{m/s} \]

### Step 3: Calculate the Uncertainty in Momentum (Δp)
The momentum (p) of an electron is given by:
\[ p = m \cdot v \]
where \( m \) is the mass of the electron. The uncertainty in momentum (Δp) can be expressed as:
\[ \Delta p = m \cdot \Delta v \]
The mass of the electron \( m \) is approximately \( 9.1 \times 10^{-31} \, \text{kg} \). Thus,
\[ \Delta p = 9.1 \times 10^{-31} \, \text{kg} \cdot 0.004 \, \text{m/s} = 3.64 \times 10^{-33} \, \text{kg m/s} \]

### Step 4: Calculate the Uncertainty in Position (Δx)
Using the uncertainty principle formula, we can rearrange it to solve for Δx:
\[ \Delta x = \frac{h}{4\pi \Delta p} \]
Substituting \( h \) (Planck's constant \( 6.626 \times 10^{-34} \, \text{J s} \)) and \( \Delta p \):
\[ \Delta x = \frac{6.626 \times 10^{-34}}{4 \cdot \pi \cdot 3.64 \times 10^{-33}} \]
Calculating this gives:
\[ \Delta x \approx \frac{6.626 \times 10^{-34}}{4 \cdot 3.14 \cdot 3.64 \times 10^{-33}} \approx 0.0001 \, \text{m} = 0.01 \, \text{cm} \]

### Final Answer
The uncertainty in locating the position of the electron is approximately:
\[ \Delta x \approx 0.01 \, \text{cm} \text{ or } 0.0001 \, \text{m} \]

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