An electron moving in a circular orbit of radius `r` makes `n` rotation per second. The magnetic field produced at the centre has magnitude

Options

A. `(mu_(0)"ne")/(2pi r)`
B. zero
C. `(mu_(0) n^(2) e)/(r)`
D. `(mu_(0)"ne")/(2r)`

Explanation

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

### Step 1: Understand the problem
We know that an electron is moving in a circular orbit of radius \( r \) and makes \( n \) rotations per second. We need to calculate the magnetic field at the center of this orbit.

### Step 2: Determine the current
The current \( I \) produced by the moving charge can be calculated using the formula:
\[ I = \frac{q}{T} \]
where \( q \) is the charge and \( T \) is the time period for one complete rotation. Since the electron makes \( n \) rotations per second, the time period \( T \) is given by:
\[ T = \frac{1}{n} \]
Thus, the current can be expressed as:
\[ I = q \cdot n \]

### Step 3: Substitute the charge of the electron
The charge of an electron \( q \) is approximately \( e = 1.6 \times 10^{-19} \) C. Therefore, the current becomes:
\[ I = n \cdot e \]

### Step 4: Use the formula for magnetic field
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, approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).

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

### Step 6: Final expression for the magnetic field
Thus, the magnitude of the magnetic field produced at the center of the circular orbit is:
\[ B = \frac{\mu_0 n e}{2r} \]

### Summary
The magnetic field produced at the center of the circular orbit of an electron moving with \( n \) rotations per second and radius \( r \) is given by:
\[ B = \frac{\mu_0 n e}{2r} \]

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