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 in terms of its angular momentum \( L \), we can follow these steps:

### Step 1: Understand the relationship between angular momentum and the electron's motion
The angular momentum \( L \) of an electron in a circular orbit can be expressed as:
\[ L = m v r \]
where \( m \) is the mass of the electron, \( v \) is its tangential velocity, and \( r \) is the radius of the orbit.

### Step 2: Write the expression for the magnetic field
The magnetic field \( B \) produced by a moving charge (the electron) at a point in space can be expressed using the Biot-Savart law. For a circular current loop, the magnetic field at the center is given by:
\[ B = \frac{\mu_0 I}{2r} \]
where \( I \) is the current due to the moving charge and \( r \) is the radius of the orbit.

### Step 3: Relate current \( I \) to the motion of the electron
The current \( I \) due to the electron can be defined as the charge passing through a point per unit time. For an electron moving in a circular path:
\[ I = \frac{q}{T} \]
where \( q \) is the charge of the electron (denoted as \( e \)), and \( T \) is the time period of one complete revolution. The time period \( T \) can be expressed as:
\[ T = \frac{2\pi r}{v} \]
Thus, the current becomes:
\[ I = \frac{e}{T} = \frac{e v}{2\pi r} \]

### Step 4: Substitute the expression for current into the magnetic field formula
Substituting \( I \) into the magnetic field expression:
\[ B = \frac{\mu_0}{2r} \cdot \frac{e v}{2\pi r} = \frac{\mu_0 e v}{4\pi r^2} \]

### Step 5: Express \( v \) in terms of \( L \)
From the angular momentum expression \( L = m v r \), we can solve for \( v \):
\[ v = \frac{L}{m r} \]
Substituting this expression for \( v \) into the magnetic field equation:
\[ B = \frac{\mu_0 e}{4\pi r^2} \cdot \frac{L}{m r} = \frac{\mu_0 e L}{4\pi m r^3} \]

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