The size of a microscopic particle 1 micron and its mass is `6 xx 10^(-13)g`. If its position may be measured to within 0.1% of its size, the uncertainity in velocity (in `cm^(-1)`) is approximately

Options

A. `(10^(-7))/(4pi)`
B. `(10^(-5))/(4pi)`
C. `10^(-5)`
D. `10^(-8)`

Answer

To solve the problem of determining the uncertainty in velocity (Δv) of a microscopic particle using the Heisenberg uncertainty principle, we can follow these steps:

### Step 1: Understand the Given Data
- Size of the particle (Δx) = 1 micron = \(1 \times 10^{-6}\) meters = \(1 \times 10^{-4}\) cm (since 1 micron = \(10^{-4}\) cm)
- Mass of the particle (m) = \(6 \times 10^{-13}\) g
- The position can be measured to within 0.1% of its size, so:
\[ \Delta x = 0.1\% \times 1 \text{ micron} = 0.001 \times 1 \times 10^{-4} \text{ cm} = 1 \times 10^{-7} \text{ cm} \]

### Step 2: Apply the Heisenberg Uncertainty Principle
The Heisenberg uncertainty principle is given by the formula:
\[ \Delta x \cdot m \cdot \Delta v \geq \frac{h}{4\pi} \]
Where:
- \(h\) (Planck's constant) = \(6.626 \times 10^{-27}\) erg·s

### Step 3: Rearranging the Formula to Find Δv
To find the uncertainty in velocity (Δv), we can rearrange the formula:
\[ \Delta v \geq \frac{h}{4\pi \cdot \Delta x \cdot m} \]

### Step 4: Substitute the Values
Substituting the known values into the equation:
- \(h = 6.626 \times 10^{-27}\) erg·s
- \(\Delta x = 1 \times 10^{-7}\) cm
- \(m = 6 \times 10^{-13}\) g

Now, substituting these values:
\[ \Delta v \geq \frac{6.626 \times 10^{-27}}{4\pi \cdot (1 \times 10^{-7}) \cdot (6 \times 10^{-13})} \]

### Step 5: Calculate the Denominator
Calculating the denominator:
\[ 4\pi \cdot (1 \times 10^{-7}) \cdot (6 \times 10^{-13}) = 4\pi \cdot 6 \times 10^{-20} \approx 75.398 \times 10^{-20} \text{ cm}^2 \text{g} \]

### Step 6: Final Calculation
Now we can calculate Δv:
\[ \Delta v \geq \frac{6.626 \times 10^{-27}}{75.398 \times 10^{-20}} \approx \frac{6.626}{75.398} \times 10^{-7} \text{ cm/s} \]

Calculating the numerical value:
\[ \Delta v \approx 0.0878 \times 10^{-7} \text{ cm/s} \approx 8.78 \times 10^{-9} \text{ cm/s} \]

### Conclusion
Thus, the uncertainty in velocity (Δv) is approximately:
\[ \Delta v \approx \frac{10^{-7}}{4\pi} \text{ cm/s} \]