Two small nuclei of mass m fuse together to form a resulting nucleus P The mass of nucleus P is (the energy released is E)

Options

A. `2m -E/c^(2)`
B. `2m + E/c^(2)`
C. `(2m)/c^(2) + E/c^(2)`
D. `(2mc^(2)-E)/c^(2)`

Explanation

To solve the problem, we need to find the mass of the resulting nucleus P after two small nuclei of mass m fuse together, releasing energy E in the process. We will use the concept of mass-energy equivalence and the principle of conservation of mass-energy.

### Step-by-Step Solution:

1. **Identify the Initial Mass:**
The initial mass before fusion consists of two small nuclei, each with mass \( m \). Therefore, the total initial mass is:
\[ \text{Initial mass} = m + m = 2m \]

2. **Understand the Energy Release:**
When the two nuclei fuse, energy \( E \) is released. According to the mass-energy equivalence principle (Einstein's equation \( E = mc^2 \)), this energy release corresponds to a mass defect \( \Delta m \).

3. **Define Mass Defect:**
The mass defect \( \Delta m \) is the difference between the initial mass and the final mass (mass of nucleus P). Thus, we can express the mass defect as:
\[ \Delta m = \text{Initial mass} - \text{Final mass} \]
Substituting the known values:
\[ \Delta m = 2m - m_P \]
where \( m_P \) is the mass of the resulting nucleus P.

4. **Relate Energy to Mass Defect:**
The energy released during the fusion can also be expressed in terms of the mass defect:
\[ E = \Delta m \cdot c^2 \]
Rearranging this gives:
\[ \Delta m = \frac{E}{c^2} \]

5. **Substitute Mass Defect into the Mass Equation:**
Now we can substitute the expression for \( \Delta m \) into our earlier equation:
\[ 2m - m_P = \frac{E}{c^2} \]

6. **Solve for the Mass of Nucleus P:**
Rearranging the equation to solve for \( m_P \):
\[ m_P = 2m - \frac{E}{c^2} \]

### Final Result:
The mass of the resulting nucleus P is:
\[ m_P = 2m - \frac{E}{c^2} \]