A particle having charge 100 times that of an electron is revolving in a circular path of radius 0.8m with one rotation per second. The magnetic field produced at the centre is

Options

A. `10^(-7) // mu_0`
B. `10^(-17) mu_0`
C. `10^(-6) mu_0 `
D. `10^(-15 ) // mu_0 `

Explanation

To find the magnetic field produced at the center of a circular path by a particle with a charge 100 times that of an electron, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the charge of the particle**:
The charge of an electron (e) is approximately \(1.6 \times 10^{-19} \, \text{C}\). Therefore, the charge (Q) of the particle is:
\[ Q = 100 \times e = 100 \times 1.6 \times 10^{-19} \, \text{C} = 1.6 \times 10^{-17} \, \text{C} \]

2. **Determine the current (I)**:
The particle makes one complete rotation per second, which means the time period (T) is 1 second. The current (I) can be calculated using the formula:
\[ I = \frac{Q}{T} = \frac{1.6 \times 10^{-17} \, \text{C}}{1 \, \text{s}} = 1.6 \times 10^{-17} \, \text{A} \]

3. **Use the formula for the magnetic field (B)**:
The magnetic field at the center of a circular loop carrying current is given by:
\[ B = \frac{\mu_0 I}{2R} \]
where \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \), and R is the radius of the circular path (0.8 m).

4. **Substituting values into the formula**:
Now substituting the values of \( I \) and \( R \):
\[ B = \frac{(4\pi \times 10^{-7}) \times (1.6 \times 10^{-17})}{2 \times 0.8} \]

5. **Calculate the denominator**:
The denominator becomes:
\[ 2 \times 0.8 = 1.6 \]

6. **Calculate the magnetic field**:
Now substituting back into the equation:
\[ B = \frac{4\pi \times 10^{-7} \times 1.6 \times 10^{-17}}{1.6} \]
The \( 1.6 \) cancels out:
\[ B = 4\pi \times 10^{-7} \times 10^{-17} = 4\pi \times 10^{-24} \, \text{T} \]

7. **Expressing the result**:
We can express the magnetic field in terms of \( \mu_0 \):
\[ B = \frac{10^{-17}}{2} \mu_0 = 10^{-17} \mu_0 \]

### Final Answer:
The magnetic field produced at the center is:
\[ B = 10^{-17} \mu_0 \, \text{T} \]