Electric field in EM waves is `E= E_0sin (kz-omegat)(hati+hatj)`, then equation of magnetic field is:

Options

A. `B=B_0(-hati+hatj) sin (kZ-omega t)`
B. `B=B_c(hati+hatj) sin (kZ-omega t)`
C. `B=B_c(-hatj+hatk) sin (kZ-omega t)`
D. `B=B_0(hati+hatj) sin (kZ-omega t)`

Answer

To find the equation of the magnetic field corresponding to the given electric field in an electromagnetic wave, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Electric Field**:
The electric field is given as:
\[ \mathbf{E} = E_0 \sin(kz - \omega t) (\hat{i} + \hat{j}) \]
This indicates that the electric field oscillates in the x-y plane.

2. **Determine the Direction of Propagation**:
The wave vector \( \mathbf{k} \) indicates the direction of wave propagation. Here, since the argument of the sine function includes \( kz \), the wave propagates in the positive z-direction.

3. **Use the Right-Hand Rule**:
In electromagnetic waves, the electric field \( \mathbf{E} \), magnetic field \( \mathbf{B} \), and the direction of wave propagation \( \mathbf{k} \) are mutually perpendicular. We can use the right-hand rule to find the direction of the magnetic field:
- Point your fingers in the direction of \( \mathbf{E} \) (which is in the x-y plane).
- Curl your fingers in the direction of wave propagation (positive z-direction).
- Your thumb will point in the direction of the magnetic field \( \mathbf{B} \).

4. **Calculate the Magnetic Field**:
Since the electric field is in the x-y plane, the magnetic field will be perpendicular to both the electric field and the direction of propagation. The magnetic field can be expressed as:
\[ \mathbf{B} = B_0 \sin(kz - \omega t) \hat{n} \]
where \( \hat{n} \) is the unit vector in the direction of \( \mathbf{B} \).

5. **Determine the Direction of \( \mathbf{B} \)**:
From the right-hand rule, if \( \mathbf{E} \) is in the direction of \( \hat{i} + \hat{j} \) (which is in the x-y plane), the magnetic field \( \mathbf{B} \) will be in the direction of \( -\hat{i} + \hat{j} \) (which is perpendicular to both \( \mathbf{E} \) and the z-direction).

6. **Express the Magnetic Field**:
Therefore, the magnetic field can be written as:
\[ \mathbf{B} = B_0 \sin(kz - \omega t) (-\hat{i} + \hat{j}) \]

7. **Final Equation**:
The final equation of the magnetic field is:
\[ \mathbf{B} = B_0 \sin(kz - \omega t) (-\hat{i} + \hat{j}) \]