The gyromagnetic ratio of an electron of charge e and mass m is equal to

Options

A. `(e^2)/(2m)`
B. `( e)/(2m^2) `
C. ` (e )/(4m)`
D. `(e )/(2m)`

Explanation

To find the gyromagnetic ratio of an electron with charge \( e \) and mass \( m \), we will follow these steps:

### Step 1: Understand the Gyromagnetic Ratio
The gyromagnetic ratio (\( \gamma \)) is defined as the ratio of the magnetic moment (\( \mu \)) to the angular momentum (\( L \)):
\[ \gamma = \frac{\mu}{L} \]

### Step 2: Calculate the Magnetic Moment
The magnetic moment (\( \mu \)) of a charged particle moving in a circular path can be expressed as:
\[ \mu = I \cdot A \]
where \( I \) is the current and \( A \) is the area.

#### Step 2.1: Find the Current
For an electron moving in a circular path of radius \( r \) with velocity \( v \), the current \( I \) can be calculated as:
\[ I = \frac{q}{T} \]
where \( q \) is the charge (which is \( e \) for an electron) and \( T \) is the time period of one complete revolution.

The time period \( T \) can be calculated as:
\[ T = \frac{2\pi r}{v} \]
Thus, the current becomes:
\[ I = \frac{e}{T} = \frac{e \cdot v}{2\pi r} \]

#### Step 2.2: Find the Area
The area \( A \) for a circular path is:
\[ A = \pi r^2 \]

#### Step 2.3: Substitute to Find Magnetic Moment
Now substituting \( I \) and \( A \) into the equation for magnetic moment:
\[ \mu = I \cdot A = \left(\frac{e \cdot v}{2\pi r}\right) \cdot (\pi r^2) = \frac{e \cdot v \cdot r}{2} \]

### Step 3: Calculate Angular Momentum
The angular momentum \( L \) of the electron moving in a circular path is given by:
\[ L = m \cdot v \cdot r \]

### Step 4: Find the Gyromagnetic Ratio
Now substituting \( \mu \) and \( L \) into the gyromagnetic ratio formula:
\[ \gamma = \frac{\mu}{L} = \frac{\frac{e \cdot v \cdot r}{2}}{m \cdot v \cdot r} \]

### Step 5: Simplify the Expression
Canceling out \( v \) and \( r \) from the numerator and denominator:
\[ \gamma = \frac{e}{2m} \]

### Conclusion
Thus, the gyromagnetic ratio of an electron is:
\[ \gamma = \frac{e}{2m} \]

### Final Answer
The correct answer is \( \frac{e}{2m} \).

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