For overcoming the Coulomb barrier for fusion, methods other than heating the fusible material have been suggested. For example, if you were to use two particle accelerators to accelerate two beams of deuterons directly toward each other so as to collide head-on, (a) what voltage would each accelerator require in order for the colliding deuterons to overcome the Coulomb barrier?

Answer

To solve the problem of determining the voltage required for two deuterons to overcome the Coulomb barrier during fusion, we will follow these steps:

### Step 1: Understand the Coulomb Barrier
The Coulomb barrier is the potential energy barrier due to electrostatic repulsion between two positively charged nuclei (in this case, deuterons). The deuterons need sufficient kinetic energy to overcome this barrier.

### Step 2: Calculate the Potential Energy at the Coulomb Barrier
The potential energy \( U \) at the point of closest approach when two deuterons are separated by a distance \( r \) is given by the formula:
\[ U = \frac{1}{4\pi \epsilon_0} \cdot \frac{q_1 \cdot q_2}{r} \]
For two deuterons, both have a charge equal to the elementary charge \( e \) (where \( e \approx 1.6 \times 10^{-19} \, \text{C} \)). Therefore, the potential energy becomes:
\[ U = \frac{1}{4\pi \epsilon_0} \cdot \frac{e^2}{2R} \]
where \( R \) is the radius of the deuteron.

### Step 3: Substitute Known Values
The radius of a deuteron is approximately \( R \approx 2.1 \times 10^{-15} \, \text{m} \). The value of \( \epsilon_0 \) (the permittivity of free space) is approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \).

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

### Step 4: Calculate the Potential Energy
Calculating the above expression gives:
\[ U \approx 2.747 \times 10^{-14} \, \text{J} \]

### Step 5: Convert Potential Energy to Electron Volts
To convert joules to electron volts, use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ U \approx \frac{2.747 \times 10^{-14}}{1.6 \times 10^{-19}} \approx 171.7 \, \text{keV} \]

### Step 6: Relate Kinetic Energy to Voltage
The kinetic energy \( K \) gained by a charge \( q \) accelerated through a potential difference \( V \) is given by:
\[ K = qV \]
For a deuteron, the charge \( q = e \). Therefore:
\[ K = eV \]
Setting \( K \) equal to the potential energy:
\[ 171.7 \, \text{keV} = (1.6 \times 10^{-19})V \]

### Step 7: Solve for Voltage
Now, rearranging the equation to solve for \( V \):
\[ V = \frac{171.7 \times 10^3 \times 1.6 \times 10^{-19}}{1.6 \times 10^{-19}} \approx 171.7 \, \text{kV} \]

### Final Answer
Thus, the voltage required for each accelerator to overcome the Coulomb barrier is approximately:
\[ V \approx 171.7 \, \text{kV} \]

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