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Step 1: Identify the Known Quantities
• Mass of the electron, $m = 9.1 \times 10^{-31}\,\text{kg}$
• Velocity of the electron, $v = 300\,\text{m s}^{-1}$
• Accuracy in velocity, $0.001\%$ (which is $0.001/100 = 1 \times 10^{-5}$ in fractional form)
• Planck’s constant, $h = 6.63 \times 10^{-34}\,\text{J s}$
• We need to find the uncertainty in position, $\Delta x$.
Step 2: Compute the Uncertainty in Velocity
The percentage uncertainty is defined by
$\displaystyle \frac{\Delta v}{v} \times 100 = 0.001.$
First, interpret $0.001$ as $0.001\% = 0.001/100$, giving $\Delta v$:
$\displaystyle 0.001 = \frac{\Delta v}{300} \times 100, \quad\text{which means}\quad \Delta v = 3 \times 10^{-3}\,\text{m s}^{-1}.$
Step 3: Use Heisenberg’s Uncertainty Principle
According to the uncertainty principle:
$\displaystyle \Delta x \, m \,\Delta v \ge \frac{h}{4\pi}.$
Rearrange to find $\Delta x$:
$\displaystyle \Delta x = \frac{h}{4\pi \, m \,\Delta v}\,.
$
Step 4: Substitute the Values
$\displaystyle
\Delta x = \frac{6.63 \times 10^{-34}}{4 \times 3.14 \times 9.1 \times 10^{-31} \times 3 \times 10^{-3}}.
$
Step 5: Perform the Calculation
Carrying out the multiplication and division:
$\displaystyle
\Delta x = 1.92 \times 10^{-2}\,\text{m}.
$
Final Answer:
The uncertainty in the position of the electron is
$\displaystyle 1.92 \times 10^{-2}\,\text{m}.$
