Electric field of an electromagnetic wave `vecE = E_0 cos (omegat-kx)hatj` . The equation of corresponding magnetic field at t=0 should be

Options

A. `vecB = E_0 sqrt(mu_0 epsilon_0) coskx hatk`
B. `vecB = (E_0 /sqrt(mu_0 epsilon_0) )coskx hatk`
C. `vecB = E_0 sqrt(mu_0 epsilon_0) coskx (-hatk)`
D. `vecB = E_0/ sqrt(mu_0 epsilon_0) coskx (-hatk)`

Answer

To find the corresponding magnetic field \( \vec{B} \) for the given electric field of an electromagnetic wave \( \vec{E} = E_0 \cos(\omega t - kx) \hat{j} \), we can follow these steps:

### Step 1: Understand the relationship between electric and magnetic fields in electromagnetic waves
In an electromagnetic wave, the electric field \( \vec{E} \) and the magnetic field \( \vec{B} \) are related by the equation:
\[ B_0 = \frac{E_0}{c} \]
where \( c \) is the speed of light, given by \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \).

### Step 2: Determine the direction of the magnetic field
Given that the electric field \( \vec{E} \) is in the \( \hat{j} \) direction (positive y-direction) and the wave propagates in the positive x-direction, we can use the right-hand rule to find the direction of the magnetic field \( \vec{B} \). According to the right-hand rule, if the thumb points in the direction of wave propagation (x-direction) and the fingers point in the direction of the electric field (y-direction), then the palm will point in the direction of the magnetic field (z-direction). Therefore, \( \vec{B} \) will be in the \( \hat{k} \) direction.

### Step 3: Write the general form of the magnetic field
The magnetic field can be expressed as:
\[ \vec{B} = B_0 \cos(\omega t - kx) \hat{k} \]
where \( B_0 \) is the amplitude of the magnetic field.

### Step 4: Substitute for \( B_0 \)
From the relationship established in Step 1, we can substitute for \( B_0 \):
\[ B_0 = \frac{E_0}{c} = E_0 \sqrt{\mu_0 \epsilon_0} \]

### Step 5: Write the complete expression for the magnetic field
Substituting \( B_0 \) into the expression for \( \vec{B} \):
\[ \vec{B} = E_0 \sqrt{\mu_0 \epsilon_0} \cos(\omega t - kx) \hat{k} \]

### Step 6: Evaluate the magnetic field at \( t = 0 \)
Now, we need to find the magnetic field at \( t = 0 \):
\[ \vec{B}(t=0) = E_0 \sqrt{\mu_0 \epsilon_0} \cos(-kx) \hat{k} \]
Using the property \( \cos(-\theta) = \cos(\theta) \):
\[ \vec{B}(t=0) = E_0 \sqrt{\mu_0 \epsilon_0} \cos(kx) \hat{k} \]

### Final Answer
Thus, the equation of the corresponding magnetic field at \( t = 0 \) is:
\[ \vec{B} = E_0 \sqrt{\mu_0 \epsilon_0} \cos(kx) \hat{k} \]