The mutual electrostatic potential energy between two protons which are at a distance of `9 xx 10^(-15)m`, in `._(92)U^(235)` nucleus is

Options

A. `1.56 xx 10^(-14) J`
B. `5.5 xx 10^(-14)J`
C. `2.56 xx 10^(-14)J`
D. `4.56 xx 10^(-14)J`

Answer

To calculate the mutual electrostatic potential energy between two protons at a distance of \(9 \times 10^{-15} \, \text{m}\), we can follow these steps:

### Step 1: Write the formula for electrostatic potential energy
The formula for the electrostatic potential energy \(U\) between two point charges is given by:
\[ U = \frac{k \cdot q_1 \cdot q_2}{r} \]
where:
- \(k\) is Coulomb's constant, approximately \(9 \times 10^9 \, \text{N m}^2/\text{C}^2\),
- \(q_1\) and \(q_2\) are the charges of the protons,
- \(r\) is the distance between the charges.

### Step 2: Identify the charges of the protons
Since both charges are protons, we have:
\[ q_1 = q_2 = q = 1.6 \times 10^{-19} \, \text{C} \]

### Step 3: Substitute the values into the formula
Substituting the values into the formula:
\[ U = \frac{9 \times 10^9 \cdot (1.6 \times 10^{-19})^2}{9 \times 10^{-15}} \]

### Step 4: Calculate \((1.6 \times 10^{-19})^2\)
Calculating the square of the charge:
\[ (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \, \text{C}^2 \]

### Step 5: Substitute and simplify
Now substituting back into the equation:
\[ U = \frac{9 \times 10^9 \cdot 2.56 \times 10^{-38}}{9 \times 10^{-15}} \]
The \(9\) in the numerator and denominator cancels out:
\[ U = \frac{10^9 \cdot 2.56 \times 10^{-38}}{10^{-15}} \]

### Step 6: Simplify the powers of ten
This simplifies to:
\[ U = 2.56 \times 10^{9 - 38 + 15} = 2.56 \times 10^{-14} \, \text{J} \]

### Final Answer
Thus, the mutual electrostatic potential energy between the two protons is:
\[ U = 2.56 \times 10^{-14} \, \text{J} \]