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 electromagnetic (EM) waves, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Electric Field**:
The electric field is given as:
\[ \mathbf{E} = E_0 \sin(kz - \omega t)(\hat{i} + \hat{j}) \]
Here, \(E_0\) is the amplitude, \(k\) is the wave number, \(\omega\) is the angular frequency, and \(\hat{i}\) and \(\hat{j}\) are the unit vectors in the x and y directions, respectively.

2. **Determine the Direction of Propagation**:
The wave is propagating in the z-direction, as indicated by the \(kz\) term in the sine function. The direction of the electric field is in the xy-plane, specifically in the direction of \(\hat{i} + \hat{j}\).

3. **Use the Right-Hand Rule**:
To find the direction of the magnetic field (\(\mathbf{B}\)), we can use the right-hand rule. According to the right-hand rule for electromagnetic waves:
- Point your thumb in the direction of the wave vector (z-direction).
- Your fingers should point in the direction of the electric field (\(\hat{i} + \hat{j}\)).
- Your palm will then face in the direction of the magnetic field.

4. **Calculate the Magnetic Field Direction**:
The electric field vector \(\mathbf{E}\) can be expressed as:
\[ \mathbf{E} = E_0 \sin(kz - \omega t) \hat{e} \]
where \(\hat{e} = \hat{i} + \hat{j}\). The magnetic field vector \(\mathbf{B}\) will be perpendicular to both \(\mathbf{E}\) and the direction of propagation (z-direction).

Since \(\hat{e}\) is in the xy-plane, the magnetic field will have components in the x and y directions. By applying the right-hand rule, we find that the magnetic field will point in the direction of \(-\hat{i} + \hat{j}\).

5. **Write the Magnetic Field Equation**:
The magnetic field can be expressed as:
\[ \mathbf{B} = B_0 \sin(kz - \omega t)(-\hat{i} + \hat{j}) \]
where \(B_0\) is the amplitude of the magnetic field.

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