An electron of charge e and mass m is moving in circular path of radius r with a uniform angular speed .` omega` Then which of the following statements are correct

Options

A. The equivalent current flowing in the circular path is proportional to `r^(2)`
B. The magnetic moment due to circular current loop is independent of m
C. The magnetic moment due to circular current loop is equal to 2e/m time the angular momentum of the electron
D. The angular momentum of the particle is proportional to the areal velocity of electron

Explanation

To solve the question regarding the electron moving in a circular path, we will analyze each statement provided and determine their validity step by step.

### Step 1: Understanding the Motion of the Electron
An electron of charge \( e \) and mass \( m \) is moving in a circular path of radius \( r \) with a uniform angular speed \( \omega \). The speed \( v \) of the electron can be expressed as:
\[ v = r \cdot \omega \]
This indicates that the electron is undergoing uniform circular motion.

**Hint:** Remember that the relationship between linear speed and angular speed is given by \( v = r \cdot \omega \).

### Step 2: Finding the Equivalent Current
The equivalent current \( I \) due to the moving charge can be calculated. The time period \( T \) for one complete revolution is:
\[ T = \frac{2\pi r}{v} = \frac{2\pi r}{r \cdot \omega} = \frac{2\pi}{\omega} \]
The current is given by:
\[ I = \frac{Q}{T} = \frac{e}{T} = \frac{e \cdot \omega}{2\pi} \]
This shows that the current is proportional to \( \omega \) and does not depend on \( r \).

**Hint:** The current is calculated by dividing the charge by the time period for one complete revolution.

### Step 3: Analyzing the Statements
1. **Statement 1:** The equivalent current flowing in the circular path is proportional to \( r^2 \).
- **Analysis:** We found that the current \( I \) is independent of \( r \). Hence, this statement is **incorrect**.

2. **Statement 2:** The magnetic moment due to a circular current loop is independent of mass \( m \).
- **Magnetic Moment Calculation:** The magnetic moment \( \mu \) is given by:
\[ \mu = I \cdot A = I \cdot \pi r^2 \]
Substituting for \( I \):
\[ \mu = \left(\frac{e \cdot \omega}{2\pi}\right) \cdot \pi r^2 = \frac{e \cdot \omega \cdot r^2}{2} \]
This shows that the magnetic moment depends on \( r^2 \) and \( \omega \), but not on \( m \). Hence, this statement is **correct**.

3. **Statement 3:** The magnetic moment due to a circular current loop is equal to \( \frac{2e}{m} \) times the angular momentum of the electron.
- **Angular Momentum Calculation:** The angular momentum \( L \) is given by:
\[ L = m \cdot v \cdot r = m \cdot (r \cdot \omega) \cdot r = m \cdot r^2 \cdot \omega \]
The relationship between magnetic moment and angular momentum is:
\[ \mu = \frac{e}{2m} L \]
This means that the magnetic moment is \( \frac{e}{2m} \) times the angular momentum, not \( \frac{2e}{m} \). Hence, this statement is **incorrect**.

4. **Statement 4:** The angular momentum of the electron is proportional to the aerial velocity of the electron.
- **Aerial Velocity:** The aerial velocity is defined as the rate at which area is swept out by the radius vector. For uniform circular motion, this is constant and can be shown to be proportional to angular momentum. Thus, this statement is **correct**.

### Conclusion
Based on the analysis:
- **Correct Statements:** 2 and 4
- **Incorrect Statements:** 1 and 3

The final answer is that statements 2 and 4 are correct.