What is the magnetic moment associated with the orbital motion of the electron if an electron of charge e moves in a circular orbit of radius r around the nucleus at a frequency v?

Options

A. `2 pi v e r ^(2)`
B. `( pi v r ^(2))/( 2)`
C. `pi v er ^(2)`
D. `( pi v e)/(r^(2))`

Explanation

To find the magnetic moment associated with the orbital motion of an electron moving in a circular orbit around a nucleus, we can follow these steps:

### Step 1: Understand the concept of magnetic moment
The magnetic moment (μ) associated with a current loop is given by the product of the current (I) flowing through the loop and the area (A) of the loop:
\[ \mu = I \cdot A \]

### Step 2: Determine the current (I)
The current (I) due to the electron moving in a circular path can be defined as the charge (e) passing through a point in the loop per unit time. Since the electron completes one full revolution in a time period (T), the frequency (ν) is related to the time period as:
\[ T = \frac{1}{\nu} \]
Thus, the current can be expressed as:
\[ I = \frac{e}{T} = e \cdot \nu \]

### Step 3: Calculate the area (A) of the circular orbit
The area (A) of a circle is given by the formula:
\[ A = \pi r^2 \]
where r is the radius of the circular orbit.

### Step 4: Substitute the values into the magnetic moment formula
Now, substituting the expressions for current (I) and area (A) into the magnetic moment formula:
\[ \mu = I \cdot A = (e \cdot \nu) \cdot (\pi r^2) \]
This simplifies to:
\[ \mu = e \cdot \nu \cdot \pi r^2 \]

### Final Expression
Thus, the magnetic moment associated with the orbital motion of the electron is:
\[ \mu = e \cdot \nu \cdot \pi r^2 \]

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