A electromagnetic wave going through a medium is given by `E = E_(0)sin (kx – omegat) and B = B_(0) sin (kx – omegat)` then

Options

A. `E_(0)k=B_(0)omega`
B. If Electric is in Z direction them magnetic field should be in -y direction.
C. Both ‘A’ and ‘B’ are correct
D. Only A is correct

Answer

To solve the problem regarding the electromagnetic wave represented by the equations \( E = E_0 \sin(kx - \omega t) \) and \( B = B_0 \sin(kx - \omega t) \), we will analyze the relationships between the electric field \( E \), magnetic field \( B \), and their respective directions.

### Step 1: Understanding the Wave Equations
The given equations represent a plane electromagnetic wave traveling through a medium. The electric field \( E \) and magnetic field \( B \) are both sinusoidal functions of space and time.

### Step 2: Direction of Propagation
For electromagnetic waves, the direction of propagation is given by the cross product of the electric field \( E \) and the magnetic field \( B \). The relationship can be expressed as:
\[ \mathbf{k} = \mathbf{E} \times \mathbf{B} \]
where \( \mathbf{k} \) is the direction of wave propagation.

### Step 3: Identifying Directions
Assuming the electric field \( E \) is in the z-direction, we can write:
\[ \mathbf{E} = E_0 \hat{z} \]
If the magnetic field \( B \) is in the y-direction, we can write:
\[ \mathbf{B} = B_0 \hat{y} \]

### Step 4: Applying the Right-Hand Rule
Using the right-hand rule:
- Point your fingers in the direction of \( \mathbf{E} \) (z-direction).
- Curl your fingers towards \( \mathbf{B} \) (y-direction).
- Your thumb will point in the direction of wave propagation (x-direction).

### Step 5: Establishing the Relationship Between E and B
In electromagnetic waves, the magnitudes of the electric and magnetic fields are related by the equation:
\[ \frac{E_0}{B_0} = c \]
where \( c \) is the speed of light in the medium.

### Step 6: Relating k and ω
The wave vector \( k \) and angular frequency \( \omega \) are related to the speed of light \( c \) as follows:
\[ c = \frac{\omega}{k} \]
From the above, we can express the relationship between the electric field and magnetic field magnitudes:
\[ E_0 = c B_0 \implies E_0 k = B_0 \omega \]

### Conclusion
Based on the analysis, we find that:
1. The relationship \( E_0 k = B_0 \omega \) holds true.
2. The directions of \( E \) and \( B \) are perpendicular to each other and to the direction of wave propagation.

Thus, both statements regarding the relationships and directions are correct.

### Final Answer
Both statements are correct: \( E_0 k = B_0 \omega \) and the directions of \( E \) and \( B \) are perpendicular to each other.

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