An electron moving in a circular orbit of radius `R` with frequency `f`. The magnetic field at the centre of the orbit is

Options

A. `(mu_0ef)/(2piR)`
B. `(mu_0ef)/(2R)`
C. `(muef^2)/(2R)`
D. zero

Explanation

To find the magnetic field at the center of a circular orbit in which an electron is moving, we can follow these steps:

### Step 1: Understand the concept of current due to moving charge
When an electron moves in a circular path, it can be treated as a current loop. The current (I) due to the moving charge can be defined as:
\[ I = \frac{q}{T} \]
where \( q \) is the charge and \( T \) is the time period of one complete revolution.

### Step 2: Relate frequency to time period
The frequency \( f \) is the reciprocal of the time period \( T \):
\[ f = \frac{1}{T} \implies T = \frac{1}{f} \]
Substituting this into the current equation gives:
\[ I = q \cdot f \]

### Step 3: Substitute the charge of the electron
The charge of an electron is denoted as \( e \). Therefore, we can write the current as:
\[ I = e \cdot f \]

### Step 4: Use the formula for magnetic field at the center of a circular loop
The magnetic field \( B \) at the center of a circular loop carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2R} \]
where \( \mu_0 \) is the permeability of free space and \( R \) is the radius of the circular path.

### Step 5: Substitute the expression for current into the magnetic field formula
Now, substituting \( I = e \cdot f \) into the magnetic field formula:
\[ B = \frac{\mu_0 (e \cdot f)}{2R} \]

### Conclusion
Thus, the magnetic field at the center of the orbit is:
\[ B = \frac{\mu_0 e f}{2R} \]