The uncertainty in the position of an electron moving with a velocity of `3xx10^(4)` cm `sec^(-1)` accurate up to 0.011 %

Options

A. 1.92 cm
B. 7.68 cm
C. 0.175 cm
D. 3.84 cm

Answer

To solve the problem of finding the uncertainty in the position of an electron moving with a given velocity, we can follow these steps:

### Step 1: Understand Heisenberg's Uncertainty Principle
Heisenberg's uncertainty principle states that the product of the uncertainties in position (Δx) and momentum (Δp) of a particle is greater than or equal to a constant:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where \( h \) is Planck's constant.

### Step 2: Identify Given Values
- Velocity of the electron, \( v = 3 \times 10^4 \, \text{cm/s} \)
- Accuracy = 0.011%
- Mass of the electron, \( m = 9.1 \times 10^{-31} \, \text{kg} \)
- Planck's constant, \( h = 6.626 \times 10^{-34} \, \text{J s} \)

### Step 3: Convert Velocity to SI Units
Convert the velocity from cm/s to m/s:
\[ v = 3 \times 10^4 \, \text{cm/s} = 3 \times 10^2 \, \text{m/s} \]

### Step 4: Calculate the Uncertainty in Velocity (Δv)
The uncertainty in velocity can be calculated using the given accuracy:
\[ \Delta v = v \times \frac{\text{accuracy}}{100} = (3 \times 10^2) \times \frac{0.011}{100} \]
\[ \Delta v = 3 \times 10^2 \times 0.00011 = 3.33 \times 10^{-2} \, \text{m/s} \]

### Step 5: Calculate the Uncertainty in Momentum (Δp)
The uncertainty in momentum (Δp) can be calculated as:
\[ \Delta p = m \cdot \Delta v = (9.1 \times 10^{-31} \, \text{kg}) \cdot (3.33 \times 10^{-2} \, \text{m/s}) \]
\[ \Delta p = 3.03 \times 10^{-32} \, \text{kg m/s} \]

### Step 6: Calculate the Uncertainty in Position (Δx)
Using Heisenberg's uncertainty principle:
\[ \Delta x \geq \frac{h}{4\pi \Delta p} \]
Substituting the values:
\[ \Delta x \geq \frac{6.626 \times 10^{-34}}{4 \cdot 3.14 \cdot 3.03 \times 10^{-32}} \]

### Step 7: Perform the Calculation
Calculating the denominator:
\[ 4 \cdot 3.14 \cdot 3.03 \times 10^{-32} \approx 3.77 \times 10^{-31} \]
Now substituting back:
\[ \Delta x \geq \frac{6.626 \times 10^{-34}}{3.77 \times 10^{-31}} \approx 0.175 \times 10^{-2} \, \text{m} \]

### Step 8: Convert to Centimeters
Convert the result from meters to centimeters:
\[ \Delta x \geq 0.175 \times 10^{-2} \, \text{m} = 0.175 \, \text{cm} \]

### Step 9: Final Answer
The uncertainty in the position of the electron is:
\[ \Delta x \geq 0.175 \, \text{cm} \]

### Conclusion
The correct answer is \( 0.175 \, \text{cm} \).

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