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: Calculate the uncertainty in velocity (ΔV)
Given that the velocity of the electron (v) is \(3 \times 10^4 \, \text{cm/s}\) and the accuracy is \(0.011\%\), we can calculate ΔV as follows:

\[ \Delta V = \frac{0.011}{100} \times v = \frac{0.011}{100} \times 3 \times 10^4 \]

Calculating this gives:

\[ \Delta V = 0.00011 \times 3 \times 10^4 = 3.3 \, \text{cm/s} \]

### Step 2: Use the Heisenberg Uncertainty Principle
According to the Heisenberg Uncertainty Principle, the uncertainty in position (Δx) is related to the uncertainty in momentum (Δp) as follows:

\[ \Delta x \geq \frac{h}{4\pi m \Delta V} \]

Where:
- \(h\) is Planck's constant, \(6.626 \times 10^{-34} \, \text{Js}\)
- \(m\) is the mass of the electron, \(9.1 \times 10^{-31} \, \text{kg}\)

### Step 3: Substitute the values into the equation
Now, substituting the values into the equation:

\[ \Delta x \geq \frac{6.626 \times 10^{-34}}{4 \times 3.14 \times 9.1 \times 10^{-31} \times 3.3} \]

Calculating the denominator:

\[ 4 \times 3.14 \times 9.1 \times 10^{-31} \times 3.3 \approx 3.7 \times 10^{-30} \]

Now substituting this back into the equation for Δx:

\[ \Delta x \geq \frac{6.626 \times 10^{-34}}{3.7 \times 10^{-30}} \approx 0.0175 \times 10^{-3} \, \text{m} \]

### Step 4: Convert meters to centimeters
Since \(1 \, \text{m} = 100 \, \text{cm}\):

\[ \Delta x \approx 0.0175 \times 10^{-3} \, \text{m} = 0.175 \, \text{cm} \]

### Conclusion
Thus, the uncertainty in the position of the electron is approximately \(0.175 \, \text{cm}\).

### Final Answer
The correct option is **0.175 cm**.

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