A particle carrying a charge equal to `100` times the charge on an electron is rotating per second in a circular path of radius `0.8 metre`. The value of the magnetic field produced at the centre will be (`mu_(0)=` permeability for vacuum)

Options

A. `(10^(-7))/(mu_(0))`
B. `10^(-17)mu_(0)`
C. `10^(-6)mu_(0)`
D. `10^(-7)mu_(0)`

Answer

To find the magnetic field produced at the center of a circular path by a charged particle, we can follow these steps:

### Step 1: Identify the charge of the particle
The charge of the particle is given as `100` times the charge of an electron. The charge of an electron (e) is approximately `1.6 x 10^-19 C`. Therefore, the charge (Q) of the particle can be calculated as:
\[ Q = 100 \times e = 100 \times 1.6 \times 10^{-19} \, C = 1.6 \times 10^{-17} \, C \]

### Step 2: Determine the current (I)
The current (I) produced by a charge moving in a circular path can be calculated using the formula:
\[ I = \frac{Q}{T} \]
where \(T\) is the time period for one complete rotation. Since the particle is rotating at `1` rotation per second, the time period \(T\) is:
\[ T = \frac{1}{1} = 1 \, s \]
Thus, the current is:
\[ I = \frac{1.6 \times 10^{-17} \, C}{1 \, s} = 1.6 \times 10^{-17} \, A \]

### Step 3: 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 can be calculated using the formula:
\[ B = \frac{\mu_0 I}{2R} \]
where \(\mu_0\) (permeability of free space) is approximately \(4\pi \times 10^{-7} \, T \cdot m/A\) and \(R\) is the radius of the circular path (0.8 m).

### Step 4: Substitute the values into the magnetic field formula
Now substituting the values into the formula:
\[ B = \frac{4\pi \times 10^{-7} \, T \cdot m/A \times 1.6 \times 10^{-17} \, A}{2 \times 0.8 \, m} \]
Calculating the denominator:
\[ 2 \times 0.8 = 1.6 \, m \]
Now substituting this back into the equation for B:
\[ B = \frac{4\pi \times 10^{-7} \times 1.6 \times 10^{-17}}{1.6} \]
The \(1.6\) in the numerator and denominator cancels out:
\[ B = 4\pi \times 10^{-7} \times 10^{-17} = 4\pi \times 10^{-24} \, T \]

### Step 5: Final calculation
The final magnetic field can be expressed as:
\[ B = 4\pi \times 10^{-24} \, T \]
This can also be represented in terms of \(\mu_0\):
\[ B = \mu_0 \times 10^{-17} \, T \]

### Conclusion
Thus, the value of the magnetic field produced at the center of the circular path is:
\[ B = 4\pi \times 10^{-24} \, T \]