To find the uncertainty in the velocity of a particle confined to a certain region, we can use the Heisenberg Uncertainty Principle, which states:
\[
\Delta x \cdot \Delta p \geq \frac{h}{4\pi}
\]
Where:
- \(\Delta x\) is the uncertainty in position,
- \(\Delta p\) is the uncertainty in momentum,
- \(h\) is Planck's constant, approximately \(6.626 \times 10^{-34} \, \text{Js}\).
### Step-by-Step Solution:
1. **Identify the given values:**
- The mass of the particle \(m = 1 \, \mu g = 1 \times 10^{-6} \, g = 1 \times 10^{-9} \, kg\).
- The region of confinement \(\Delta x = 1 \, mm = 1 \times 10^{-3} \, m\).
2. **Use the uncertainty principle:**
According to the uncertainty principle:
\[
\Delta x \cdot \Delta p \geq \frac{h}{4\pi}
\]
We can express momentum \(p\) as \(p = mv\), so the uncertainty in momentum \(\Delta p\) can be expressed as:
\[
\Delta p = m \cdot \Delta v
\]
3. **Substituting into the uncertainty equation:**
Now substituting \(\Delta p\) into the uncertainty principle:
\[
\Delta x \cdot (m \cdot \Delta v) \geq \frac{h}{4\pi}
\]
4. **Rearranging to find \(\Delta v\):**
Rearranging the equation gives:
\[
\Delta v \geq \frac{h}{4\pi \Delta x m}
\]
5. **Substituting the known values:**
Now substitute \(h = 6.626 \times 10^{-34} \, Js\), \(\Delta x = 1 \times 10^{-3} \, m\), and \(m = 1 \times 10^{-9} \, kg\):
\[
\Delta v \geq \frac{6.626 \times 10^{-34}}{4\pi \cdot (1 \times 10^{-3}) \cdot (1 \times 10^{-9})}
\]
6. **Calculating the right side:**
First, calculate \(4\pi \cdot (1 \times 10^{-3}) \cdot (1 \times 10^{-9})\):
\[
4\pi \cdot (1 \times 10^{-3}) \cdot (1 \times 10^{-9}) \approx 1.25664 \times 10^{-12}
\]
Now calculate:
\[
\Delta v \geq \frac{6.626 \times 10^{-34}}{1.25664 \times 10^{-12}} \approx 5.27 \times 10^{-21} \, m/s
\]
7. **Converting to cm/s:**
To convert \(\Delta v\) from meters per second to centimeters per second:
\[
5.27 \times 10^{-21} \, m/s = 5.27 \times 10^{-21} \times 100 \, cm/s = 5.27 \times 10^{-19} \, cm/s
\]
8. **Final answer:**
The uncertainty in the velocity of the particle is:
\[
\Delta v \approx 5.27 \times 10^{-21} \, cm/s
\]
