It is not possible to determine preciselt both the position and momentum (or velocity) of a small moving particle such as electron, proton etc. This is known as Heisenber uncertainty principle. The mathemactical form of this principle is :
`Delta x.Delta p ge (h)/(4pi)` (constant)
However this principle is irrevalent in case of bigger particles such as a cup, ball, car etc., that we come across in our daily life.
Given that the mass of electron is `9.1 xx 10^(-31) kg` and velocity of electron is `2.2 xx 10^(6) ms^(-1)`, if uncertainty in its velocity is `0.1%`, the uncertainty in position would be

Options

A. `26nm`
B. `32nm`
C. `48 nm`
D. `50 nm`

Explanation

To solve the problem, we need to calculate the uncertainty in position (Δx) of an electron given the uncertainty in its velocity (Δv). We will use the Heisenberg Uncertainty Principle, which states:

\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]

Where:
- \( \Delta x \) = uncertainty in position
- \( \Delta p \) = uncertainty in momentum
- \( h \) = Planck's constant (\(6.63 \times 10^{-34} \, \text{Js}\))

### Step-by-Step Solution:

1. **Calculate the uncertainty in velocity (Δv)**:
Given that the velocity of the electron (v) is \(2.2 \times 10^6 \, \text{ms}^{-1}\) and the uncertainty in velocity is \(0.1\%\):
\[ \Delta v = 0.1\% \text{ of } v = \frac{0.1}{100} \times 2.2 \times 10^6 = 2.2 \times 10^3 \, \text{ms}^{-1} \]

2. **Calculate the uncertainty in momentum (Δp)**:
The uncertainty in momentum is given by:
\[ \Delta p = m \cdot \Delta v \]
Where \(m\) is the mass of the electron, \(9.1 \times 10^{-31} \, \text{kg}\):
\[ \Delta p = 9.1 \times 10^{-31} \, \text{kg} \cdot 2.2 \times 10^3 \, \text{ms}^{-1} = 2.002 \times 10^{-27} \, \text{kg m/s} \]

3. **Use the Heisenberg Uncertainty Principle to find Δx**:
Rearranging the uncertainty principle formula:
\[ \Delta x \geq \frac{h}{4\pi \Delta p} \]
Substituting the values:
\[ \Delta x \geq \frac{6.63 \times 10^{-34} \, \text{Js}}{4 \cdot 3.14 \cdot 2.002 \times 10^{-27} \, \text{kg m/s}} \]

4. **Calculate Δx**:
First, calculate the denominator:
\[ 4 \cdot 3.14 \cdot 2.002 \times 10^{-27} \approx 2.507 \times 10^{-26} \]
Now, calculate Δx:
\[ \Delta x \geq \frac{6.63 \times 10^{-34}}{2.507 \times 10^{-26}} \approx 2.64 \times 10^{-8} \, \text{m} = 26.4 \, \text{nm} \]

### Final Answer:
The uncertainty in position (Δx) is approximately \(26.4 \, \text{nm}\).

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