Due to orbital motion, magnetic moment is:

Options

A. `(1)/(4)` evr
B. `(1)/(2)` evr
C. evr
D. `(ev)/(2r)`

Explanation

To determine the magnetic moment due to the orbital motion of an electron, we can follow these steps:

### Step 1: Understand the Concept of Current
When an electron moves in a circular orbit, it creates a current. The current \( I \) can be defined as the charge passing through a point in a circuit per unit time.

### Step 2: Calculate the Time Period of the Electron's Orbit
The time \( T \) taken for one complete revolution of the electron in a circular orbit of radius \( r \) with velocity \( v \) is given by:
\[ T = \frac{2\pi r}{v} \]

### Step 3: Calculate the Current
The current \( I \) due to the electron's motion can be expressed as:
\[ I = \frac{e}{T} \]
Substituting the expression for \( T \):
\[ I = \frac{e}{\frac{2\pi r}{v}} = \frac{ev}{2\pi r} \]

### Step 4: Determine the Area of the Orbit
The area \( A \) of the circular orbit is given by:
\[ A = \pi r^2 \]

### Step 5: Calculate the Magnetic Moment
The magnetic moment \( m \) (or magnetic dipole moment) is given by the product of the current and the area:
\[ m = I \cdot A \]
Substituting the expressions for \( I \) and \( A \):
\[ m = \left(\frac{ev}{2\pi r}\right) \cdot (\pi r^2) = \frac{evr}{2} \]

### Step 6: Final Expression for Magnetic Moment
Thus, the magnetic moment due to the orbital motion of the electron can be expressed as:
\[ m = \frac{evr}{2} \]