The orbital speed of an electron orbiting around the nucleus in a circular orbit of radius r is v. Then the magnetic dipole moment of the electron will be

Options

A. ever
B. `(evr)/(2)`
C. `(ev)/(2r)`
D. `(vr)/(2e)`

Answer

To find the magnetic dipole moment of an electron orbiting around the nucleus in a circular orbit of radius \( r \) with orbital speed \( v \), we can follow these steps:

### Step 1: Understand the concept of magnetic dipole moment
The magnetic dipole moment (\( \mu \)) of a current-carrying loop is given by the formula:
\[ \mu = I \cdot A \]
where \( I \) is the current and \( A \) is the area of the loop.

### Step 2: Determine the current due to the electron's motion
The current \( I \) can be defined as the charge passing through a point per unit time. For an electron moving in a circular orbit:
\[ I = \frac{Q}{T} \]
where \( Q \) is the charge of the electron (denoted as \( e \)) and \( T \) is the time period for one complete revolution.

### Step 3: Calculate the time period \( T \)
The time period \( T \) can be calculated using the formula for the circumference of the circular path divided by the speed:
\[ T = \frac{2\pi r}{v} \]

### Step 4: Substitute \( T \) into the current formula
Substituting \( T \) into the current formula gives:
\[ I = \frac{e}{T} = \frac{e}{\frac{2\pi r}{v}} = \frac{e \cdot v}{2\pi r} \]

### Step 5: Calculate the area \( A \) of the circular orbit
The area \( A \) of the circular orbit is given by:
\[ A = \pi r^2 \]

### Step 6: Substitute \( I \) and \( A \) into the magnetic dipole moment formula
Now, substituting \( I \) and \( A \) into the magnetic dipole moment formula:
\[ \mu = I \cdot A = \left(\frac{e \cdot v}{2\pi r}\right) \cdot \left(\pi r^2\right) \]

### Step 7: Simplify the expression
Simplifying the expression:
\[ \mu = \frac{e \cdot v}{2\pi r} \cdot \pi r^2 = \frac{e \cdot v \cdot r}{2} \]

### Conclusion
Thus, the magnetic dipole moment of the electron is:
\[ \mu = \frac{e \cdot v \cdot r}{2} \]

### Final Answer
The magnetic dipole moment of the electron is \( \frac{e \cdot v \cdot r}{2} \).

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