Which of the following is correct regarding electromagnetic wave?

Options

A. In electromagnetic wave, electirc and magnetic field vector oscillate perpendicular to direction of progation
B. If wave is propagating in `x`-direction `E = E_(0)sin omega(t-(x)/(c ))`
`B = B_(0)sin omega (t-(x)/(c ))`,
`E_(0) = cB_(0),c = (1)/(sqrt(mu_(0)epsilon_(0)))`
C. Intensity of the wave `I = 1//2epsilon_(0)E_(0)^(2)c`
D. All options are correct

Answer

To determine which statements regarding electromagnetic waves are correct, let's analyze the properties of electromagnetic waves step by step.

### Step-by-Step Solution:

1. **Understanding Electromagnetic Waves**:
Electromagnetic waves consist of oscillating electric (E) and magnetic (B) fields that propagate through space. These fields oscillate perpendicular to each other and to the direction of wave propagation.

2. **Direction of Propagation**:
If we consider an electromagnetic wave propagating in the x-direction, the electric field vector can be oriented in the y-direction and the magnetic field vector in the z-direction. This means:
- Electric field (E) oscillates in the y-direction.
- Magnetic field (B) oscillates in the z-direction.

3. **Transverse Nature**:
Electromagnetic waves are transverse waves. This means that the oscillations of the electric and magnetic fields are perpendicular to the direction of wave propagation. Thus, the motion of the wave is perpendicular to the motion of the particles (in this case, the electric and magnetic fields).

4. **Mutual Perpendicularity**:
Since the electric field is in the y-direction and the magnetic field is in the z-direction while the wave propagates in the x-direction, we confirm that these vectors are mutually perpendicular.

5. **Equations of the Fields**:
The equations governing the electric and magnetic fields in an electromagnetic wave can be described mathematically. The electric field can be expressed as:
\[ E(x, t) = E_0 \sin(kx - \omega t) \]
and the magnetic field as:
\[ B(x, t) = B_0 \sin(kx - \omega t) \]
where \(E_0\) and \(B_0\) are the amplitudes, \(k\) is the wave number, and \(\omega\) is the angular frequency.

6. **Relationship Between Electric and Magnetic Fields**:
The relationship between the magnitudes of the electric and magnetic fields in an electromagnetic wave is given by:
\[ c = \frac{E}{B} \]
where \(c\) is the speed of light in vacuum.

7. **Intensity of the Wave**:
The intensity \(I\) of an electromagnetic wave can be expressed as:
\[ I = \frac{1}{2} \epsilon_0 E_0^2 c \]
where \(\epsilon_0\) is the permittivity of free space.

### Conclusion:
Based on the analysis, we can conclude that:
- The electric and magnetic fields oscillate perpendicular to the direction of propagation (Correct).
- The electric field vector varies with time (Correct).
- The relationship between the fields and the speed of light is valid (Correct).
- The formula for intensity is also correct.

Thus, all options regarding electromagnetic waves are correct.