Suppose two deuterons must get as close as `10^(-14) m` in order for the nuclear force to overcome the repulsive electrostatic force. The height of the electrostatic harrier is nearest to

Options

A. `0.14 M e V`
B. `2.3 M e V`
C. `1.8 xx 10 M e V`
D. `0.56 M e V`

Explanation

To find the height of the electrostatic barrier when two deuterons come close to each other, we can use the formula for the potential energy due to electrostatic repulsion. The formula is given by:

\[ U = \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r} \]

Where:
- \( U \) is the potential energy (height of the barrier),
- \( \epsilon_0 \) is the permittivity of free space (\( \approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)),
- \( e \) is the charge of an electron (\( \approx 1.6 \times 10^{-19} \, \text{C} \)),
- \( r \) is the distance at which the nuclear force can overcome the electrostatic force (given as \( 10^{-14} \, \text{m} \)).

### Step 1: Substitute the values into the formula

Using the given values:
- \( r = 10^{-14} \, \text{m} \)
- \( e = 1.6 \times 10^{-19} \, \text{C} \)
- \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)

The potential energy \( U \) becomes:

\[ U = \frac{1}{4 \pi (8.85 \times 10^{-12})} \cdot \frac{(1.6 \times 10^{-19})^2}{10^{-14}} \]

### Step 2: Calculate \( \frac{(1.6 \times 10^{-19})^2}{10^{-14}} \)

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

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

Now, substituting this back into the equation:

\[ U = \frac{1}{4 \pi (8.85 \times 10^{-12})} \cdot \frac{2.56 \times 10^{-38}}{10^{-14}} \]

This simplifies to:

\[ U = \frac{1}{4 \pi (8.85 \times 10^{-12})} \cdot 2.56 \times 10^{-24} \]

### Step 3: Calculate \( \frac{1}{4 \pi (8.85 \times 10^{-12})} \)

Calculating \( 4 \pi (8.85 \times 10^{-12}) \):

\[ 4 \pi \approx 12.5664 \]

Thus,

\[ 4 \pi (8.85 \times 10^{-12}) \approx 1.112 \times 10^{-10} \]

Now, calculating \( \frac{1}{1.112 \times 10^{-10}} \):

\[ \frac{1}{1.112 \times 10^{-10}} \approx 8.99 \times 10^{9} \]

### Step 4: Final Calculation

Now substituting back into the equation for \( U \):

\[ U \approx (8.99 \times 10^{9}) \cdot (2.56 \times 10^{-24}) \]

Calculating this gives:

\[ U \approx 2.30 \times 10^{-14} \, \text{J} \]

### Step 5: Convert Joules to Electron Volts

To convert Joules to electron volts, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):

\[ U \approx \frac{2.30 \times 10^{-14}}{1.6 \times 10^{-19}} \approx 1.44 \times 10^{5} \, \text{eV} \]

### Step 6: Convert to Mega Electron Volts

Finally, converting to Mega Electron Volts:

\[ U \approx 0.144 \, \text{MeV} \]

Thus, the height of the electrostatic barrier is approximately \( 0.14 \, \text{MeV} \).

### Final Answer
The height of the electrostatic barrier is nearest to **0.14 MeV**.