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Step-by-Step Solution
Step 1: Identify the given data
• Speed of electron, $v = 600\text{ m/s}$
• Accuracy in speed = $0.005\%$
• Planck’s constant, $h = 6.6 \times 10^{-34}\,\text{J·s} \,(\text{or kg·m}^2\text{/s})$ (approx.)
• Mass of electron, $m = 9.1 \times 10^{-31}\,\text{kg}$
• We need to find the uncertainty in position, $\Delta x$.
Step 2: Calculate the uncertainty in velocity, $\Delta v$
The percentage error in velocity is given as:
$ \displaystyle \frac{\Delta v}{v} \times 100 = 0.005 \% $
Substitute $v = 600 \,\text{m/s}$:
$ \displaystyle 0.005 = \frac{\Delta v}{600} \times 100 $
Rearranging for $\Delta v$:
$ \displaystyle \Delta v = 600 \times \frac{0.005}{100} = 600 \times 0.00005 = 3 \times 10^{-2}\,\text{m/s} $
Step 3: Apply Heisenberg’s Uncertainty Principle
Heisenberg's uncertainty principle in one dimension is stated as:
$ \displaystyle \Delta x \times m \Delta v \;\ge\; \frac{h}{4 \pi} $
We solve for $\Delta x$ to get:
$ \displaystyle \Delta x = \frac{h}{4 \pi \, m \,\Delta v} $
Step 4: Substitute the numerical values
Using the values:
$ \displaystyle \Delta x
= \frac{6.6 \times 10^{-34}}{4 \times \pi \times (9.1 \times 10^{-31}) \times (3 \times 10^{-2})} $
Evaluate step by step:
Denominator part: $4 \times \pi \approx 4 \times 3.14 = 12.56.$
Then, $12.56 \times 9.1 \times 10^{-31} \approx 114.296 \times 10^{-31} \approx 1.14296 \times 10^{-29}.$
Next, multiply by $3 \times 10^{-2}$: $1.14296 \times 10^{-29} \times 3 \times 10^{-2} \approx 3.42888 \times 10^{-31}.$
Now, divide $6.6 \times 10^{-34}$ by $3.42888 \times 10^{-31}$:
$ \displaystyle \Delta x \approx 1.92 \times 10^{-3}\,\text{m} $
Step 5: State the final answer
Therefore, the uncertainty in the position ($\Delta x$) of the electron is
$ \displaystyle 1.92 \times 10^{-3}\,\text{m}. $
