The angular momentum of electron is J. if e=charge of electron , m= mass of electron , then its magnetic moment will be

Options

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

Explanation

To find the magnetic moment (μ) of an electron in terms of its angular momentum (J), we can follow these steps:

### Step 1: Understand the relationship between current and charge
The current (I) due to the moving electron can be expressed as the charge (e) passing through a point in time (T). Thus, we have:
\[ I = \frac{e}{T} \]

### Step 2: Determine the time period (T) for one complete revolution
For an electron moving in a circular orbit, the time period (T) can be calculated as:
\[ T = \frac{2\pi r}{v} \]
where \( r \) is the radius of the orbit and \( v \) is the speed of the electron.

### Step 3: Substitute T into the current equation
Substituting the expression for T into the current equation gives:
\[ I = \frac{e}{\frac{2\pi r}{v}} = \frac{ev}{2\pi r} \]

### Step 4: Calculate the area (A) of the circular orbit
The area (A) of the circular orbit can be given by:
\[ A = \pi r^2 \]

### Step 5: Relate magnetic moment (μ) to current (I) and area (A)
The magnetic moment (μ) can be expressed as:
\[ \mu = I \cdot A \]
Substituting the expressions for I and A, we get:
\[ \mu = \left(\frac{ev}{2\pi r}\right) \cdot (\pi r^2) = \frac{evr}{2} \]

### Step 6: Relate linear momentum to angular momentum
The angular momentum (J) of the electron can be expressed as:
\[ J = mvr \]
where \( m \) is the mass of the electron. We can solve for \( vr \):
\[ vr = \frac{J}{m} \]

### Step 7: Substitute vr into the magnetic moment equation
Now substituting \( vr \) into the magnetic moment equation gives:
\[ \mu = \frac{e}{2} \cdot \frac{J}{m} \]
Thus, we have:
\[ \mu = \frac{eJ}{2m} \]

### Final Answer
The magnetic moment (μ) of the electron in terms of its angular momentum (J) is:
\[ \mu = \frac{eJ}{2m} \]

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