Problem 31 - Physics for Scientists and Engineers with Modern Physics

Question

(III) Calculate the maximum kinetic energy of the electron when a muon decays from rest via \(\mu^{-} \rightarrow \mathrm{e}^{-}+\overline{v}_{\mathrm{e}}+\nu_{\mu}\) [Hint. In what direction do the two neutrinos move relative to the electron in order to give the electron the maximum kinetic energy? Both energy and momentum are conserved; use relativistic formulas.]

Step-by-Step Solution

Answer

The maximum kinetic energy of the electron is approximately 105.15 MeV.

Step 1: Understand the Problem

We need to find the maximum kinetic energy of the electron when a muon decays from rest into an electron and two neutrinos. The hint suggests considering the direction of the neutrinos relative to the electron to maximize its kinetic energy.

Step 2: Conservation of Energy

When the muon decays, the total energy must be conserved. The initial energy of the muon at rest is its rest energy, given by \(E_\mu = m_\mu c^2\). In the decay, the total energy is shared among the electron, the electron neutrino, and the muon neutrino.

Step 3: Conservation of Momentum

Since the initial muon is at rest, its initial momentum is zero. Therefore, momentum must also be conserved after the decay. The momenta of the electron and the two neutrinos must sum to zero.

Step 4: Finding Maximum Electron Kinetic Energy Stance

To maximize the kinetic energy of the electron, the two neutrinos should move in the same direction, opposite to the electron, ensuring the electron takes as much kinetic energy as possible while still conserving momentum.

Step 5: Relativistic Energy of Electron

Use the energy-momentum relation for the electron: \(E_e^2 = (p_ec)^2 + (m_ec^2)^2\), where \(E_e\) is the total energy of the electron, and \(p_e\) is the momentum of the electron. Maximize \(K_e = E_e - m_ec^2\).

Step 6: Applying Energy and Momentum Conservation Equations

Use conserved energy: \(E_\mu = E_e + E_{\bar{u}_e} + E_{u_\mu}\), and conserved momentum: \(p_e + p_{\bar{u}_e} + p_{u_\mu} = 0\). Use these to solve for the maximum \(E_e\) and thus the maximum \(K_e\).

Step 7: Express Kinetic Energy

Kinetic energy \(K_e\) is maximum when the electron takes energy difference, which is \(m_\mu c^2 - m_ec^2\) minus a small energy portion carried by neutrinos, almost making \(K_e \approx m_\mu c^2 - m_ec^2\).

Step 8: Calculate Maximum Kinetic Energy

Using the approximation that neutrinos take negligible energy compared to the rest mass energy difference, \(K_{e_{max}}\approx m_\mu c^2 - m_ec^2\). Substitute known values of \(m_\mu\approx 105.66 \text{ MeV/c}^2\) and \(m_e\approx 0.511 \text{ MeV/c}^2\).

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. In the decay of a muon, we are interested in the kinetic energy of the resulting electron. The kinetic energy of the electron can be expressed as \( K_e = E_e - m_e c^2 \), where \( E_e \) is the total energy of the electron and \( m_e c^2 \) is its rest energy.

To achieve maximum kinetic energy for the electron, it should carry as much as possible of the total energy available from the decay. Practically, this means that the direction and speed of the electron, relative to the neutrinos produced in the decay, are essential considerations. By applying the principles of energy conservation and momentum conservation, we can determine the conditions under which the electron achieves its highest possible kinetic energy.

Energy Conservation

The principle of energy conservation states that the total energy in an isolated system remains constant over time.

In the case of muon decay, the total initial energy is just the rest energy of the muon, given by \( E_\mu = m_\mu c^2 \). After the decay, this energy is distributed among the electron, the electron neutrino, and the muon neutrino. Thus, the energy conservation equation can be written as:

• \( E_\mu = E_e + E_{\overline{v}_e} + E_{u_\mu} \)

To find the maximum kinetic energy of the electron, we ideally want a situation where the neutrinos carry away the least amount of energy. This ensures that the electron receives most of the energy, respecting the law of energy conservation.

Momentum Conservation

Momentum conservation is another fundamental principle, stating that the total momentum of a system remains constant unless acted upon by an external force.

Since the muon is initially at rest, its total momentum is zero. After decay, the momentum of the electron and the two neutrinos must sum up to zero to conserve momentum:

• \( p_e + p_{\overline{v}_e} + p_{u_\mu} = 0 \)

To maximize the kinetic energy of the electron, the two neutrinos should move in the same direction, opposite to the electron. This configuration allows the electron to gain the most significant momentum in one direction, resulting in the highest energy and momentum transfer to the electron within the constraints of momentum conservation.

Relativistic Equations

Relativistic equations become essential when particles, like electrons in muon decay, move at speeds close to the speed of light. These equations help us understand how energy and momentum are related at relativistic speeds.

The key equation here is the energy-momentum relation: \[ E_e^2 = (p_ec)^2 + (m_ec^2)^2 \] where \( E_e \) is the total energy, \( p_e \) is the momentum of the electron, and \( m_e \) its rest mass.

Using this formula, we can derive the maximum kinetic energy by expressing \( K_e = E_e - m_ec^2 \). This equation links the electron's kinetic energy to its momentum and highlights the complex interactions involved when dealing with particles in a relativistic framework. Understanding these interactions is crucial for accurately calculating particle energies and momenta in high-speed decay events.