Calculate the impact parameter of a 5 MeV particle scattered by `90^(@)` when it approaches a gold nucleus.

Options

A. `15.27 x 10^(-14)` m
B. `23.23 xx 10^(-14)` m
C. `2.27 xx 10^(-14)`m
D. `77.2 xx 10^(-14)`m

Answer

To calculate the impact parameter of a 5 MeV particle scattered by \(90^\circ\) when it approaches a gold nucleus, we can follow these steps:

### Step 1: Understand the Problem
The impact parameter \(b\) is the perpendicular distance from the center of the nucleus to the path of the incoming particle if it were to continue in a straight line. The particle is an alpha particle with an energy of 5 MeV, and it scatters at an angle of \(90^\circ\).

### Step 2: Relevant Formula
The impact parameter can be calculated using the formula:
\[ b = \frac{Z_1 Z_2 e^2 \cot(\theta/2)}{4 \pi \epsilon_0 E} \]
where:
- \(Z_1\) is the atomic number of the alpha particle (which is 2 for helium),
- \(Z_2\) is the atomic number of the gold nucleus (which is 79),
- \(e\) is the charge of the electron (\(1.6 \times 10^{-19} \, \text{C}\)),
- \(\epsilon_0\) is the permittivity of free space (\(8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2\)),
- \(E\) is the energy of the alpha particle in joules,
- \(\theta\) is the scattering angle.

### Step 3: Convert Energy to Joules
The energy of the alpha particle is given as 5 MeV. To convert this to joules:
\[ E = 5 \, \text{MeV} = 5 \times 1.6 \times 10^{-13} \, \text{J} = 8 \times 10^{-13} \, \text{J} \]

### Step 4: Calculate \(\cot(\theta/2)\)
For a scattering angle of \(90^\circ\):
\[ \theta = 90^\circ \implies \cot(\theta/2) = \cot(45^\circ) = 1 \]

### Step 5: Substitute Values into the Formula
Now substituting the values into the impact parameter formula:
\[ b = \frac{(2)(79)(1.6 \times 10^{-19})^2 \cdot 1}{4 \pi (8.85 \times 10^{-12})(8 \times 10^{-13})} \]

### Step 6: Calculate the Numerator
Calculating the numerator:
\[ (2)(79)(1.6 \times 10^{-19})^2 = 2 \times 79 \times 2.56 \times 10^{-38} = 404.48 \times 10^{-38} = 4.0448 \times 10^{-36} \]

### Step 7: Calculate the Denominator
Calculating the denominator:
\[ 4 \pi (8.85 \times 10^{-12})(8 \times 10^{-13}) \approx 4 \times 3.14 \times 8.85 \times 10^{-12} \times 8 \times 10^{-13} \approx 8.84 \times 10^{-24} \]

### Step 8: Final Calculation
Now, substituting the values back into the impact parameter equation:
\[ b = \frac{4.0448 \times 10^{-36}}{8.84 \times 10^{-24}} \approx 4.57 \times 10^{-13} \, \text{m} \]

### Step 9: Conclusion
Thus, the impact parameter \(b\) is approximately \(4.57 \times 10^{-13} \, \text{m}\).