An electromagnetic wave is propagating in vacuum along z-axis, the electric field component is given by `E_(x)=E_(0)sin (kz-omegat)`. Then magnetic components is

Options

A. `B_(x)=(E_(0))/(C)sin (kz-omegat)`
B. `B_(y)=(B_(0))/(C)sin (kz-omegat)`
C. `B_(y)=(E_(0))/(C)sin (kz-omegat)`
D. `B_(y)=B_(0)Csin (kz-omegat)`

Answer

To find the magnetic field component of an electromagnetic wave propagating in vacuum along the z-axis, we can follow these steps:

### Step 1: Identify the given electric field
The electric field component is given as:
\[ E_x = E_0 \sin(kz - \omega t) \]

### Step 2: Understand the relationship between electric and magnetic fields in electromagnetic waves
In a vacuum, the electric field \( \vec{E} \) and magnetic field \( \vec{B} \) are related by the speed of light \( c \):
\[ c = \frac{E_0}{B_0} \]
where \( E_0 \) is the amplitude of the electric field and \( B_0 \) is the amplitude of the magnetic field.

### Step 3: Calculate the amplitude of the magnetic field
From the relationship above, we can express the amplitude of the magnetic field as:
\[ B_0 = \frac{E_0}{c} \]

### Step 4: Determine the direction of the magnetic field
Since the wave is propagating along the z-axis and the electric field is in the x-direction, we can use the right-hand rule to find the direction of the magnetic field. According to the right-hand rule:
- Point your thumb in the direction of wave propagation (z-axis).
- Your fingers point in the direction of the electric field (x-axis).
- The palm will face in the direction of the magnetic field (y-axis).

Thus, the magnetic field \( \vec{B} \) will be in the y-direction.

### Step 5: Write the equation for the magnetic field
Since the magnetic field is in the y-direction and has the same sinusoidal form as the electric field, we can write:
\[ B_y = B_0 \sin(kz - \omega t) \]

Substituting \( B_0 \) from Step 3:
\[ B_y = \frac{E_0}{c} \sin(kz - \omega t) \]

### Final Answer
The magnetic field component is:
\[ B_y = \frac{E_0}{c} \sin(kz - \omega t) \]