In a Geiger-Marsden experiment, an `alpha`-particle of kinetic energy 4 MeV is scattered by `8^@` while approaching towards gold (Z =79) nucleus. Calculate impact parameter.

Answer

To solve the problem of calculating the impact parameter in a Geiger-Marsden experiment where an alpha particle is scattered by a gold nucleus, we will follow these steps:

### Step 1: Gather Given Data
- Kinetic energy of the alpha particle, \( E = 4 \text{ MeV} \)
- Scattering angle, \( \theta = 8^\circ \)
- Atomic number of gold, \( Z = 79 \)

### Step 2: Convert Kinetic Energy to Joules
The kinetic energy in joules can be calculated using the conversion factor \( 1 \text{ MeV} = 1.6 \times 10^{-13} \text{ Joules} \).

\[ E = 4 \text{ MeV} = 4 \times 1.6 \times 10^{-13} \text{ Joules} = 6.4 \times 10^{-13} \text{ Joules} \]

### Step 3: Understand the Formula for Impact Parameter
The impact parameter \( b \) can be calculated using the formula:

\[ b = \frac{Z e^2 \cot(\theta/2)}{E} \cdot \frac{1}{4 \pi \epsilon_0} \]

Where:
- \( e \) is the charge of an electron, \( e = 1.6 \times 10^{-19} \text{ C} \)
- \( \epsilon_0 \) is the permittivity of free space, \( \epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{N m}^2 \)

### Step 4: Calculate \( \cot(\theta/2) \)
First, we need to calculate \( \theta/2 \):

\[ \theta/2 = \frac{8^\circ}{2} = 4^\circ \]

Now, calculate \( \cot(4^\circ) \):

\[ \cot(4^\circ) = \frac{1}{\tan(4^\circ)} \approx 14.3 \quad (\text{using a calculator}) \]

### Step 5: Substitute Values into the Formula
Substituting the values into the impact parameter formula:

\[ b = \frac{79 \times (1.6 \times 10^{-19})^2 \times 14.3}{6.4 \times 10^{-13}} \cdot \frac{1}{4 \pi (8.85 \times 10^{-12})} \]

Calculating \( (1.6 \times 10^{-19})^2 \):

\[ (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \]

Now substituting everything back into the equation:

\[ b = \frac{79 \times 2.56 \times 10^{-38} \times 14.3}{6.4 \times 10^{-13} \times 4 \pi (8.85 \times 10^{-12})} \]

### Step 6: Calculate the Denominator
Calculating the denominator \( 6.4 \times 10^{-13} \times 4 \pi (8.85 \times 10^{-12}) \):

\[ 4 \pi (8.85 \times 10^{-12}) \approx 1.11 \times 10^{-10} \]

Thus,

\[ 6.4 \times 10^{-13} \times 1.11 \times 10^{-10} \approx 7.1 \times 10^{-23} \]

### Step 7: Final Calculation
Now substituting back into the equation for \( b \):

\[ b \approx \frac{79 \times 2.56 \times 10^{-38} \times 14.3}{7.1 \times 10^{-23}} \]

Calculating the numerator:

\[ 79 \times 2.56 \times 10^{-38} \times 14.3 \approx 2.88 \times 10^{-36} \]

Thus,

\[ b \approx \frac{2.88 \times 10^{-36}}{7.1 \times 10^{-23}} \approx 4.07 \times 10^{-13} \text{ m} \]

### Final Answer
The impact parameter \( b \) is approximately:

\[ b \approx 4.07 \times 10^{-13} \text{ m} \]