Calculate the equivalent energy of electrons and proton at rest. Given that mass of electron `=9.1xx10^(-31)kg` and mass of proton `=1.673xx10^(-27)kg`.

Explanation

To calculate the equivalent energy of an electron and a proton at rest, we will use the mass-energy equivalence formula given by Einstein's theory of relativity:

\[ E = mc^2 \]

where:
- \( E \) is the energy,
- \( m \) is the mass,
- \( c \) is the speed of light in vacuum (\( c \approx 3 \times 10^8 \) m/s).

### Step 1: Calculate the energy of the electron

1. **Identify the mass of the electron:**
\[ m_e = 9.1 \times 10^{-31} \text{ kg} \]

2. **Substitute the values into the energy formula:**
\[ E_e = m_e c^2 = (9.1 \times 10^{-31} \text{ kg}) \times (3 \times 10^8 \text{ m/s})^2 \]

3. **Calculate \( c^2 \):**
\[ c^2 = (3 \times 10^8)^2 = 9 \times 10^{16} \text{ m}^2/\text{s}^2 \]

4. **Calculate the energy in joules:**
\[ E_e = 9.1 \times 10^{-31} \times 9 \times 10^{16} = 8.19 \times 10^{-14} \text{ J} \]

5. **Convert joules to mega electron volts (MeV):**
- Use the conversion factor \( 1 \text{ MeV} = 1.6 \times 10^{-13} \text{ J} \):
\[ E_e = \frac{8.19 \times 10^{-14} \text{ J}}{1.6 \times 10^{-13} \text{ J/MeV}} \approx 0.511 \text{ MeV} \]

### Step 2: Calculate the energy of the proton

1. **Identify the mass of the proton:**
\[ m_p = 1.673 \times 10^{-27} \text{ kg} \]

2. **Substitute the values into the energy formula:**
\[ E_p = m_p c^2 = (1.673 \times 10^{-27} \text{ kg}) \times (3 \times 10^8 \text{ m/s})^2 \]

3. **Calculate the energy in joules:**
\[ E_p = 1.673 \times 10^{-27} \times 9 \times 10^{16} = 1.5057 \times 10^{-10} \text{ J} \]

4. **Convert joules to mega electron volts (MeV):**
\[ E_p = \frac{1.5057 \times 10^{-10} \text{ J}}{1.6 \times 10^{-13} \text{ J/MeV}} \approx 941.1 \text{ MeV} \]

### Final Results

- The equivalent energy of the electron at rest is approximately **0.511 MeV**.
- The equivalent energy of the proton at rest is approximately **941.1 MeV**.