When an electron revolves around the nucleus, then the ratio of magnetic moment to angular momentum is

Options

A. `e/(2m)`
B. `(2e)/m`
C. `e/m`
D. `(e/m)^2`

Answer

To solve the problem of finding the ratio of magnetic moment to angular momentum for an electron revolving around a nucleus, we will follow these steps:

### Step 1: Define Magnetic Moment
The magnetic moment (M) of a current loop is given by the formula:
\[ M = I \cdot A \]
where \( I \) is the current and \( A \) is the area of the loop.

### Step 2: Calculate the Area
For an electron revolving in a circular orbit of radius \( r \), the area \( A \) can be calculated as:
\[ A = \pi r^2 \]

### Step 3: Determine the Current
The current \( I \) due to the revolving electron can be defined as the charge passing through a point per unit time. The charge of the electron is \( e \). The time \( T \) taken to complete one revolution is the circumference of the circle divided by the velocity \( v \):
\[ T = \frac{2\pi r}{v} \]
Thus, the current \( I \) is:
\[ I = \frac{Q}{T} = \frac{e}{T} = \frac{e \cdot v}{2\pi r} \]

### Step 4: Substitute Current in Magnetic Moment Formula
Now we can substitute the expression for current \( I \) into the magnetic moment formula:
\[ M = I \cdot A = \left(\frac{e \cdot v}{2\pi r}\right) \cdot (\pi r^2) \]
This simplifies to:
\[ M = \frac{e \cdot v \cdot r}{2} \]

### Step 5: Define Angular Momentum
The angular momentum \( L \) of the electron is given by:
\[ L = m \cdot v \cdot r \]
where \( m \) is the mass of the electron.

### Step 6: Calculate the Ratio of Magnetic Moment to Angular Momentum
Now we can find the ratio of magnetic moment \( M \) to angular momentum \( L \):
\[ \text{Ratio} = \frac{M}{L} = \frac{\frac{e \cdot v \cdot r}{2}}{m \cdot v \cdot r} \]
Here, \( v \) and \( r \) cancel out:
\[ \text{Ratio} = \frac{e}{2m} \]

### Conclusion
The ratio of magnetic moment to angular momentum for an electron revolving around a nucleus is:
\[ \frac{M}{L} = \frac{e}{2m} \]