Consider an electron obrbiting the nucleus with speed `v` in an orbit of radius `r`. The ratio of the magetic moment to the orbtial angular momentum of the electron is independent of:

Options

A. radius `r`
B. speed `v`
C. charge of electron `e`
D. mass of electron `m_(e )`

Answer

To solve the problem, we need to find the ratio of the magnetic moment to the orbital angular momentum of an electron orbiting a nucleus.

### Step-by-Step Solution:

1. **Understanding Magnetic Moment**:
The magnetic moment (μ) of an electron in orbit can be expressed using the formula:
\[ \mu = \frac{e \cdot h}{4 \pi} \]
where \( e \) is the charge of the electron and \( h \) is Planck's constant.

2. **Understanding Orbital Angular Momentum**:
The orbital angular momentum (L) of the electron can be expressed using the formula:
\[ L = m \cdot v \cdot r \]
where \( m \) is the mass of the electron, \( v \) is its velocity, and \( r \) is the radius of the orbit.

3. **Finding the Ratio**:
We need to find the ratio of the magnetic moment to the orbital angular momentum:
\[ \text{Ratio} = \frac{\mu}{L} \]
Substituting the expressions for μ and L:
\[ \text{Ratio} = \frac{\frac{e \cdot h}{4 \pi}}{m \cdot v \cdot r} \]

4. **Simplifying the Ratio**:
This simplifies to:
\[ \text{Ratio} = \frac{e \cdot h}{4 \pi \cdot m \cdot v \cdot r} \]
Now, we can analyze the components of this ratio.

5. **Identifying Independence**:
- The constants \( e \) (charge of the electron), \( h \) (Planck's constant), and \( m \) (mass of the electron) are constants.
- The ratio depends on \( v \) (speed of the electron) and \( r \) (radius of the orbit).

6. **Conclusion**:
The ratio of the magnetic moment to the orbital angular momentum is independent of:
- The radius \( r \) of the orbit.
- The speed \( v \) of the electron.
- It is dependent on the charge \( e \) of the electron and the mass \( m \) of the electron.

### Final Answer:
The ratio of the magnetic moment to the orbital angular momentum of the electron is independent of the radius \( r \) and the speed \( v \).