Calculate the height of the Coulomb barrier for the head on collision of two deuterons, with effective radius 2.1 fm.

Answer

To calculate the height of the Coulomb barrier for the head-on collision of two deuterons with an effective radius of 2.1 femtometers (fm), we can follow these steps:

### Step 1: Understand the Concept of Coulomb Barrier
The Coulomb barrier is the potential energy barrier due to the electrostatic repulsion between two positively charged particles. In this case, we have two deuterons, each with a charge of \( +e \) (where \( e = 1.6 \times 10^{-19} \) C).

### Step 2: Use the Formula for Potential Energy
The potential energy \( U \) between two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by:
\[ U = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r} \]
For two deuterons, \( q_1 = q_2 = e \), and the distance \( r \) when they are just touching each other is \( 2R \), where \( R \) is the effective radius of a deuteron.

### Step 3: Substitute the Values
Given:
- \( R = 2.1 \, \text{fm} = 2.1 \times 10^{-15} \, \text{m} \)
- \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)
- \( e = 1.6 \times 10^{-19} \, \text{C} \)

Substituting these into the potential energy formula:
\[ U = \frac{1}{4 \pi (8.85 \times 10^{-12})} \frac{(1.6 \times 10^{-19})^2}{2 \times (2.1 \times 10^{-15})} \]

### Step 4: Calculate the Potential Energy
Calculating the denominator:
\[ 2R = 2 \times 2.1 \times 10^{-15} = 4.2 \times 10^{-15} \, \text{m} \]

Now substituting into the potential energy formula:
\[ U = \frac{(1.6 \times 10^{-19})^2}{4 \pi (8.85 \times 10^{-12}) \times (4.2 \times 10^{-15})} \]

### Step 5: Perform the Calculation
1. Calculate \( (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \, \text{C}^2 \)
2. Calculate \( 4 \pi (8.85 \times 10^{-12}) \approx 1.11 \times 10^{-10} \)
3. Now substitute these values into the equation for \( U \):
\[ U \approx \frac{2.56 \times 10^{-38}}{1.11 \times 10^{-10} \times 4.2 \times 10^{-15}} \approx \frac{2.56 \times 10^{-38}}{4.66 \times 10^{-25}} \approx 5.49 \times 10^{-14} \, \text{J} \]

### Step 6: Final Result
The height of the Coulomb barrier for the head-on collision of two deuterons is approximately:
\[ U \approx 5.49 \times 10^{-14} \, \text{J} \]