The uncertainty in the position of an electron `(mass = 9.1 xx 10^-28 g)` moving with a velocity of `3.0 xx 10^4 cm s^-1` accurate up to `0.001 %` will be
(Use `(h)/(4 pi)` in the uncertainty expression, where `h = 6.626 xx 10^-27 erg - s`)

Options

A. 1.93 cm
B. 3.84 cm
C. 5.76 cm
D. 7.68 cm

Explanation

To solve the problem of finding the uncertainty in the position of an electron, we will use the Heisenberg Uncertainty Principle, which states that:

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

Where:
- \(\Delta x\) = uncertainty in position
- \(\Delta p\) = uncertainty in momentum
- \(h\) = Planck's constant

### Step-by-Step Solution:

**Step 1: Calculate the uncertainty in velocity (\(\Delta v\))**

The velocity of the electron is given as \(3.0 \times 10^4 \, \text{cm/s}\) with an accuracy of \(0.001\%\).

To find the uncertainty in velocity, we calculate:

\[ \Delta v = \text{Accuracy} \times \text{Velocity} \]

\[ \Delta v = \frac{0.001}{100} \times 3.0 \times 10^4 \, \text{cm/s} = 0.3 \, \text{cm/s} \]

**Step 2: Calculate the uncertainty in momentum (\(\Delta p\))**

The uncertainty in momentum is given by:

\[ \Delta p = m \cdot \Delta v \]

Where \(m\) is the mass of the electron, given as \(9.1 \times 10^{-28} \, \text{g}\).

Now substituting the values:

\[ \Delta p = 9.1 \times 10^{-28} \, \text{g} \cdot 0.3 \, \text{cm/s} = 2.73 \times 10^{-28} \, \text{g cm/s} \]

**Step 3: Substitute into the Heisenberg Uncertainty Principle**

Now we can substitute \(\Delta p\) into the Heisenberg Uncertainty Principle equation:

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

We know \(h = 6.626 \times 10^{-27} \, \text{erg s}\), and \(4\pi \approx 12.566\).

Calculating \(\frac{h}{4\pi}\):

\[ \frac{h}{4\pi} = \frac{6.626 \times 10^{-27}}{12.566} \approx 5.276 \times 10^{-28} \, \text{erg s} \]

**Step 4: Solve for \(\Delta x\)**

Now substituting \(\Delta p\) into the equation:

\[ \Delta x \cdot 2.73 \times 10^{-28} \geq 5.276 \times 10^{-28} \]

Solving for \(\Delta x\):

\[ \Delta x \geq \frac{5.276 \times 10^{-28}}{2.73 \times 10^{-28}} \approx 1.93 \, \text{cm} \]

### Final Answer:

The uncertainty in the position of the electron is approximately \(1.93 \, \text{cm}\).

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