The muon has the same change as an electron but a mass that is `207` times greater. The negetively charged muon can bind to a prtoton to form a new type of hydrogen atom. How does the binding energy `E_(Bmu)`, of the muon the ground state of muonic hydrogen atom campare with the binding energy `E_(Be')` of an electron in the ground state of a conventional hydrogen atom?

Options

A. a.`|E_(Bmu)| = |E_(Be)|`
B. b.`|E_(Bmu)|~~ 100 |E_(Be)|`
C. c.`|E_(Bmu)|~~|E_(Be)|//107`
D. d.`|E_(Bmu)|=207|E_(Be)|`

Explanation

To compare the binding energy of a muon in the ground state of a muonic hydrogen atom with the binding energy of an electron in the ground state of a conventional hydrogen atom, we can follow these steps:

### Step 1: Understanding Binding Energy Formula
The binding energy \( E_B \) for a particle in a hydrogen-like atom can be expressed as:
\[ E_B = -\frac{m e^4}{2 (4 \pi \epsilon_0)^2 \hbar^2 n^2} \]
Where:
- \( m \) is the mass of the particle (electron or muon),
- \( e \) is the charge of the particle,
- \( \epsilon_0 \) is the permittivity of free space,
- \( \hbar \) is the reduced Planck's constant,
- \( n \) is the principal quantum number (for ground state, \( n = 1 \)).

### Step 2: Binding Energy of Electron in Hydrogen Atom
For the electron in a conventional hydrogen atom, the binding energy \( E_{B_e} \) can be written as:
\[ E_{B_e} = -\frac{m_e e^4}{2 (4 \pi \epsilon_0)^2 \hbar^2} \]
Where \( m_e \) is the mass of the electron.

### Step 3: Binding Energy of Muon in Muonic Hydrogen Atom
For the muon, which has a mass \( m_\mu \) that is 207 times greater than the mass of the electron, the binding energy \( E_{B_\mu} \) can be expressed as:
\[ E_{B_\mu} = -\frac{m_\mu e^4}{2 (4 \pi \epsilon_0)^2 \hbar^2} \]
Where \( m_\mu = 207 m_e \).

### Step 4: Ratio of Binding Energies
To find the ratio of the binding energies, we can write:
\[ \frac{E_{B_\mu}}{E_{B_e}} = \frac{m_\mu}{m_e} = \frac{207 m_e}{m_e} = 207 \]

### Step 5: Conclusion
This means that the binding energy of the muon in the ground state of the muonic hydrogen atom is 207 times greater than the binding energy of the electron in the ground state of a conventional hydrogen atom:
\[ E_{B_\mu} = 207 E_{B_e} \]

### Final Answer
The binding energy \( E_{B_\mu} \) of the muon in the ground state of the muonic hydrogen atom is 207 times greater than the binding energy \( E_{B_e} \) of the electron in the ground state of a conventional hydrogen atom.

---