A particle is moving wirh velocity `vecv=hati+3hatj` and it produces an electric field at a point given by `vecE=2hatk`. It will produce magnetic field at that point equal to (all quantities are in SI units)

Options

A. `(6hati-2hatj)mu_0epsilon_0`
B. `(6hati+2hatj)mu_0epsilon_0`
C. zero
D. cannot be determine

Explanation

To solve the problem of finding the magnetic field produced by a particle moving with a given velocity and producing an electric field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Quantities**:
- Velocity of the particle: \(\vec{v} = \hat{i} + 3\hat{j}\)
- Electric field at the point: \(\vec{E} = 2\hat{k}\)

2. **Understand the Relationship Between Electric Field and Magnetic Field**:
- The magnetic field \(\vec{B}\) produced by a moving charge can be calculated using the formula:
\[ \vec{B} = \frac{\mu_0}{4\pi} \cdot \frac{Q \cdot \vec{v} \times \hat{R}}{R^2} \]
- Here, \(\hat{R}\) is the unit vector pointing from the charge to the point where the field is being calculated, and \(R\) is the distance from the charge to that point.

3. **Cross Product Calculation**:
- The electric field is given as \(\vec{E} = 2\hat{k}\), which indicates that the charge is producing an electric field in the z-direction.
- To find the magnetic field, we need to compute the cross product \(\vec{v} \times \hat{R}\).
- Assuming \(\hat{R} = \hat{k}\) (since the electric field is in the z-direction), we can calculate:
\[ \vec{v} \times \hat{R} = (\hat{i} + 3\hat{j}) \times \hat{k} \]

4. **Perform the Cross Product**:
- Using the determinant method for the cross product:
\[ \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 3 & 0 \\ 0 & 0 & 1 \end{vmatrix} \]
- This results in:
\[ \hat{i}(3 \cdot 1 - 0 \cdot 0) - \hat{j}(1 \cdot 1 - 0 \cdot 0) + \hat{k}(0 \cdot 0 - 0 \cdot 3) = 3\hat{i} - 1\hat{j} \]

5. **Substituting Back into the Magnetic Field Equation**:
- Now substituting back into the magnetic field equation:
\[ \vec{B} = \frac{\mu_0}{4\pi} \cdot Q \cdot (3\hat{i} - 1\hat{j}) / R^2 \]
- Since we are not given the charge \(Q\) or the distance \(R\), we can express the magnetic field in terms of these variables:
\[ \vec{B} = \frac{\mu_0}{4\pi} \cdot Q \cdot (3\hat{i} - 1\hat{j}) \cdot \frac{1}{R^2} \]

6. **Final Result**:
- The magnetic field at the point is given by:
\[ \vec{B} = \mu_0 \cdot (3\hat{i} - \hat{j}) \quad \text{(assuming Q and R are constants)} \]

### Final Answer:
The magnetic field produced at that point is:
\[ \vec{B} = \mu_0 (3\hat{i} - \hat{j}) \]