If an electron is moving with velocity v in an orbit of radius r then the equivalent magnetic field at the centre will be:

Options

A. (a)`(mu_0 ev)/(r^2)`
B. (b)`(mu_0 ev)/(2r)`
C. (c)`(mu_0 ev)/(4 pi r^2)`
D. (d)`(mu_0 e)/(2r)`

Explanation

To find the equivalent magnetic field at the center of an orbit where an electron is moving with velocity \( v \) and radius \( r \), we can follow these steps:

### Step 1: Understand the concept of current due to the moving electron
When an electron moves in a circular path, it creates a current. The current \( I \) can be defined as the charge \( Q \) passing through a point in a given time \( T \).

### Step 2: Calculate the time period \( T \) for one complete revolution
The time \( T \) taken for the electron to complete one full revolution in a circular path of radius \( r \) with velocity \( v \) is given by:
\[ T = \frac{2\pi r}{v} \]

### Step 3: Calculate the current \( I \)
The current \( I \) produced by the moving electron can be calculated using the formula:
\[ I = \frac{Q}{T} \]
Here, the charge \( Q \) of the electron is denoted as \( e \). Thus, substituting for \( T \):
\[ I = \frac{e}{T} = \frac{e}{\frac{2\pi r}{v}} = \frac{ev}{2\pi r} \]

### Step 4: Use the formula for the 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:
\[ B = \frac{\mu_0 I}{2r} \]
where \( \mu_0 \) is the permeability of free space.

### Step 5: Substitute the expression for current \( I \) into the magnetic field formula
Substituting \( I = \frac{ev}{2\pi r} \) into the equation for \( B \):
\[ B = \frac{\mu_0 \left(\frac{ev}{2\pi r}\right)}{2r} = \frac{\mu_0 ev}{4\pi r^2} \]

### Conclusion
Thus, the equivalent magnetic field at the center of the orbit is:
\[ B = \frac{\mu_0 ev}{4\pi r^2} \]