A point charge is moving in a circular path of radius `10cm` with angular frequency `40pi` ra//s. The magnetic field produced by it at the center is `3.8xx10^(-10)T`. Then the value of charge is:

Options

A. `3muC`
B. `4muC`
C. `1muC`
D. `9muC`

Answer

To find the value of the charge \( Q \) moving in a circular path, we can use the relationship between the magnetic field produced by a moving charge and its parameters. Here’s a step-by-step solution:

### Step 1: Understand the given parameters
- Radius of the circular path, \( R = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Angular frequency, \( \omega = 40\pi \, \text{rad/s} \)
- Magnetic field at the center, \( B = 3.8 \times 10^{-10} \, \text{T} \)

### Step 2: Relate current to charge and frequency
The current \( I \) due to a charge \( Q \) moving in a circular path can be expressed as:
\[ I = Q \cdot f \]
where \( f \) is the frequency. The frequency can be related to angular frequency by:
\[ f = \frac{\omega}{2\pi} \]

### Step 3: Substitute the frequency into the current equation
Substituting for \( f \):
\[ I = Q \cdot \frac{\omega}{2\pi} \]

### Step 4: Use the formula for the magnetic field
The magnetic field \( B \) at the center of a circular loop carrying current \( I \) is given by:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{I}{R^2} \]
where \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \).

### Step 5: Substitute \( I \) into the magnetic field equation
Substituting \( I \) into the magnetic field equation:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{Q \cdot \frac{\omega}{2\pi}}{R^2} \]

### Step 6: Rearranging the equation to solve for \( Q \)
Rearranging gives:
\[ Q = \frac{B \cdot 4\pi R^2}{\mu_0 \cdot \frac{\omega}{2\pi}} \]
This simplifies to:
\[ Q = \frac{B \cdot 8\pi^2 R^2}{\mu_0 \cdot \omega} \]

### Step 7: Substitute the known values
Now substitute \( B = 3.8 \times 10^{-10} \, \text{T} \), \( R = 0.1 \, \text{m} \), \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \), and \( \omega = 40\pi \):
\[ Q = \frac{(3.8 \times 10^{-10}) \cdot 8\pi^2 (0.1)^2}{(4\pi \times 10^{-7}) \cdot (40\pi)} \]

### Step 8: Calculate the value of \( Q \)
Calculating the constants:
\[ Q = \frac{(3.8 \times 10^{-10}) \cdot 8\pi^2 (0.01)}{(4\pi \times 10^{-7}) \cdot (40\pi)} \]
\[ = \frac{(3.8 \times 10^{-10}) \cdot 8 \cdot \pi}{(4 \times 40) \times 10^{-7}} \]
\[ = \frac{(3.8 \times 10^{-10}) \cdot 8 \cdot \pi}{160 \times 10^{-7}} \]
\[ = \frac{3.8 \times 8 \cdot \pi}{160} \times 10^{-3} \]

Calculating the numerical values:
\[ = \frac{30.4 \cdot \pi}{160} \times 10^{-3} \approx 3 \times 10^{-6} \, \text{C} \]

Thus, the value of the charge \( Q \) is approximately:
\[ Q \approx 3 \, \mu C \]

### Final Answer:
The value of the charge is \( 3 \, \mu C \).

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