If the anti-neutrino had a mass of `3eV//c^(2)`where is the speed of light) instead of mass, what should be the range of the kinetic energy. K of the electron? .

Options

A. `0leKle0.8xx10^(6) eV`
B. `3.0eVleKle0.8xx10^(6) eV`
C. `3.0eVleKlt0.8xx10^(6) eV`
D. `0leKlt0.8xx10^(6) eV`

Answer

To solve the problem, we need to determine the range of the kinetic energy (K) of the electron when the anti-neutrino has a mass of \(3 \, \text{eV}/c^2\).

### Step 1: Understand the energy conservation in the decay process
In a decay process involving an electron and an anti-neutrino, the total energy is conserved. The maximum energy (E_max) carried by the electron can be expressed as:

\[ E_{\text{max}} = K + E_{\text{anti-neutrino}} \]

where:
- \(K\) is the kinetic energy of the electron,
- \(E_{\text{anti-neutrino}} = m_{\text{anti-neutrino}} c^2\).

### Step 2: Calculate the energy of the anti-neutrino
Given that the mass of the anti-neutrino is \(3 \, \text{eV}/c^2\), we can calculate its energy:

\[ E_{\text{anti-neutrino}} = m_{\text{anti-neutrino}} c^2 = 3 \, \text{eV} \]

### Step 3: Calculate the maximum energy of the electron
The rest mass energy of the electron (\(m_e c^2\)) is approximately \(0.511 \, \text{MeV}\) or \(511 \, \text{eV}\). Therefore, we can express the maximum energy of the electron as:

\[ E_{\text{max}} = 511 \, \text{eV} \]

### Step 4: Set up the equation for kinetic energy
Using the conservation of energy, we can substitute the values we have:

\[ 511 \, \text{eV} = K + 3 \, \text{eV} \]

### Step 5: Solve for the kinetic energy \(K\)
Rearranging the equation gives us:

\[ K = 511 \, \text{eV} - 3 \, \text{eV} = 508 \, \text{eV} \]

### Step 6: Determine the range of kinetic energy
Since the kinetic energy of the electron must be non-negative, we can conclude that:

\[ 0 \leq K < 508 \, \text{eV} \]

Thus, the range of kinetic energy \(K\) of the electron is:

\[ 0 \leq K < 508 \, \text{eV} \]

### Summary of the Solution
The range of the kinetic energy of the electron when the anti-neutrino has a mass of \(3 \, \text{eV}/c^2\) is:

\[ 0 \leq K < 508 \, \text{eV} \]