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 equation of the corresponding magnetic field for the given electric field of an electromagnetic wave, we can follow these steps:

### Step 1: Identify the Given Electric Field
The electric field of the electromagnetic wave is given by:
\[ \vec{E} = E_0 \cos(\omega t - kx) \hat{j} \]
This indicates that the electric field oscillates in the y-direction (since it is multiplied by \(\hat{j}\)).

### Step 2: Understand the Propagation Direction
In electromagnetic waves, the electric field \(\vec{E}\), magnetic field \(\vec{B}\), and the direction of wave propagation are mutually perpendicular. The wave is propagating in the positive x-direction (as indicated by the term \(-kx\)).

### Step 3: Use the Relationship Between Electric and Magnetic Fields
The relationship between the magnitudes of the electric field \(E_0\) and the magnetic field \(B_0\) in an electromagnetic wave is given by:
\[ B_0 = \frac{E_0}{c} \]
where \(c\) is the speed of light in vacuum, given by:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]

### Step 4: Write the Magnetic Field Equation
The magnetic field \(\vec{B}\) can be expressed as:
\[ \vec{B} = B_0 \cos(\omega t - kx) \hat{n} \]
where \(\hat{n}\) is the unit vector in the direction of the magnetic field.

### Step 5: Determine the Direction of the Magnetic Field
Since the electric field is in the y-direction (\(\hat{j}\)), and the wave is propagating in the x-direction (\(\hat{i}\)), the magnetic field must be in the z-direction (\(\hat{k}\)). This is determined using the right-hand rule, where:
\[ \vec{E} \times \vec{B} \propto \vec{k} \]
Thus, the magnetic field will be in the direction of \(\hat{k}\).

### Step 6: Substitute the Magnitude of the Magnetic Field
Now we can substitute \(B_0\) into the equation:
\[ B_0 = \frac{E_0}{c} = E_0 \sqrt{\mu_0 \epsilon_0} \]
Thus, the magnetic field can be expressed as:
\[ \vec{B} = \frac{E_0}{c} \cos(\omega t - kx) \hat{k} \]

### Step 7: Finalize the Magnetic Field Equation
Substituting \(c\) into the equation, we get:
\[ \vec{B} = \frac{E_0}{\frac{1}{\sqrt{\mu_0 \epsilon_0}}} \cos(\omega t - kx) \hat{k} = E_0 \sqrt{\mu_0 \epsilon_0} \cos(\omega t - kx) \hat{k} \]

### Conclusion
The equation of the corresponding magnetic field at \(t = 0\) is:
\[ \vec{B} = E_0 \sqrt{\mu_0 \epsilon_0} \cos(-kx) \hat{k} \]
or simply:
\[ \vec{B} = E_0 \sqrt{\mu_0 \epsilon_0} \cos(kx) \hat{k} \]