An electron orbiting around a nucleus has angular momentum L. The magnetic field produced by the electron at the centre of the orbit can be expressed as :

Options

A. `B = (mu_(0) e//8pi mr^(3))L`
B. `B =(mu_(0)e//4 pi mr^(3))L`
C. `B = (mu_(0) e//pi mr^(3))L`
D. `B =(e//4 pi epsilon_(0) mr^(3))L`

Explanation

To find the magnetic field produced by an electron orbiting around a nucleus, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Angular Momentum**:
The angular momentum \( L \) of an electron in circular motion can be expressed as:
\[ L = mvr \]
where \( m \) is the mass of the electron, \( v \) is its velocity, and \( r \) is the radius of the orbit.

2. **Relate Current to Angular Momentum**:
The current \( I \) produced by the moving charge (the electron) can be defined as:
\[ I = \frac{q}{T} \]
where \( q \) is the charge of the electron and \( T \) is the period of revolution. The period \( T \) can be expressed in terms of the velocity and radius:
\[ T = \frac{2\pi r}{v} \]
Thus, we can rewrite the current as:
\[ I = \frac{qv}{2\pi r} \]

3. **Substitute Current into Magnetic Field Formula**:
The magnetic field \( B \) at the center of the orbit can be calculated using the formula:
\[ B = \frac{\mu_0 I}{2r} \]
Substituting the expression for current \( I \):
\[ B = \frac{\mu_0}{2r} \cdot \frac{qv}{2\pi r} = \frac{\mu_0 qv}{4\pi r^2} \]

4. **Express Velocity in Terms of Angular Momentum**:
From the angular momentum expression \( L = mvr \), we can express \( v \) as:
\[ v = \frac{L}{mr} \]
Substituting this back into the magnetic field equation gives:
\[ B = \frac{\mu_0 q}{4\pi r^2} \cdot \frac{L}{mr} = \frac{\mu_0 q L}{4\pi m r^3} \]

5. **Final Expression for Magnetic Field**:
Thus, the magnetic field produced by the electron at the center of its orbit can be expressed as:
\[ B = \frac{\mu_0 q L}{4\pi m r^3} \]

6. **Identify the Correct Option**:
Given the options, we can see that the expression matches with option B:
\[ B = \frac{\mu_0 e L}{4\pi m r^3} \]
where \( e \) is the charge of the electron.

### Conclusion:
The magnetic field produced by the electron at the center of its orbit can be expressed as:
\[ B = \frac{\mu_0 e L}{4\pi m r^3} \]