If uncertainty principle is applied to an object of mass 1 miligram, the product of uncertainly in velocity and position will be:

Options

A. `0.2xx10^(-4)m^(2)s^(-1)`
B. `0.52xx10^(6)m^(2)s^(-1)`
C. `0.52xx10^(-28)m^(2)s^(-1)`
D. `2xx10^(-34)m^(2)s^(-1)`

Explanation

To solve the problem of finding the product of uncertainty in position (\( \Delta x \)) and uncertainty in velocity (\( \Delta v \)) for an object of mass 1 milligram using the Heisenberg Uncertainty Principle, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Heisenberg Uncertainty Principle**:
The principle states that the product of the uncertainties in position and momentum is greater than or equal to a constant. Mathematically, it is expressed as:
\[ \Delta x \cdot \Delta v \geq \frac{h}{4\pi m} \]
where:
- \( \Delta x \) is the uncertainty in position,
- \( \Delta v \) is the uncertainty in velocity,
- \( h \) is the Planck constant,
- \( m \) is the mass of the object.

2. **Convert Mass to SI Units**:
The mass given is 1 milligram. We need to convert this into kilograms:
\[ 1 \text{ mg} = 1 \times 10^{-3} \text{ g} = 1 \times 10^{-6} \text{ kg} \]

3. **Use the Value of Planck's Constant**:
The value of Planck's constant \( h \) is:
\[ h = 6.626 \times 10^{-34} \text{ J s} \]

4. **Substitute Values into the Uncertainty Principle**:
We can substitute the values into the uncertainty principle formula:
\[ \Delta x \cdot \Delta v \geq \frac{6.626 \times 10^{-34}}{4 \pi (1 \times 10^{-6})} \]

5. **Calculate the Denominator**:
First, calculate \( 4 \pi (1 \times 10^{-6}) \):
\[ 4 \pi \approx 4 \times 3.14 \approx 12.56 \]
Thus,
\[ 4 \pi (1 \times 10^{-6}) \approx 12.56 \times 10^{-6} \]

6. **Calculate the Product**:
Now, substitute this back into the equation:
\[ \Delta x \cdot \Delta v \geq \frac{6.626 \times 10^{-34}}{12.56 \times 10^{-6}} \]
Performing the division:
\[ \Delta x \cdot \Delta v \geq \frac{6.626}{12.56} \times 10^{-34 + 6} = 0.528 \times 10^{-28} \text{ m}^2/\text{s} \]

7. **Final Answer**:
Rounding this value gives:
\[ \Delta x \cdot \Delta v \geq 0.52 \times 10^{-28} \text{ m}^2/\text{s} \]

### Conclusion
The product of the uncertainty in position and velocity for an object of mass 1 milligram is approximately \( 0.52 \times 10^{-28} \text{ m}^2/\text{s} \).