A plane electromagnetic wave of wavelength `lambda` has an intensity I. It is propagating along the positive Y-direction. The allowed expression for the electric and magnetic fields ar given by :
`vecE=sqrt((2I)/(in_(0)c))cos[(2pi)/(lambda)(y-ct)]hatk`,

Options

A. `vecB=+(1)/(c)E " veci`
`vecE=sqrt((2I)/(in_(0)c))cos[(2pi)/(lambda)(y+ct)]hatk`,
B. `vecB=(1)/(c)E " veci`
`vecE=sqrt((I)/(in_(0)c))cos[(2pi)/(lambda)(y-ct)]hatk`,
C. `vecB=(1)/(c)E " veci`
`vecE=sqrt((I)/(in_(0)c))cos[(2pi)/(lambda)(y-ct)]hati`,
D. `vecB=(1)/(c)E " veci`

Answer

To solve the problem of finding the expressions for the electric field (E) and magnetic field (B) of a plane electromagnetic wave propagating along the positive Y-direction with a given intensity (I), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Direction of Propagation**:
The wave is propagating in the positive Y-direction. For electromagnetic waves, the electric field (E) and magnetic field (B) are perpendicular to each other and to the direction of propagation.

2. **Identifying the Orientation of E and B**:
Since the wave is traveling in the Y-direction, we can assume:
- The electric field (E) can be oriented along the X-direction.
- The magnetic field (B) can be oriented along the Z-direction.
This configuration satisfies the right-hand rule: if you point your thumb in the direction of propagation (Y), your index finger in the direction of E (X), your middle finger will point in the direction of B (Z).

3. **Using the Given Expression for Electric Field**:
The expression for the electric field is given as:
\[ \vec{E} = \sqrt{\frac{2I}{\epsilon_0 c}} \cos\left(\frac{2\pi}{\lambda}(y - ct)\right) \hat{k} \]
Here, \(\hat{k}\) indicates that the electric field is in the Z-direction.

4. **Finding the Magnetic Field**:
The relationship between the electric field (E) and the magnetic field (B) in electromagnetic waves is given by:
\[ B = \frac{E}{c} \]
Therefore, substituting for E:
\[ \vec{B} = \frac{1}{c} \sqrt{\frac{2I}{\epsilon_0 c}} \cos\left(\frac{2\pi}{\lambda}(y - ct)\right) \hat{i} \]
Here, \(\hat{i}\) indicates that the magnetic field is in the X-direction.

5. **Final Expressions**:
Thus, the expressions for the electric and magnetic fields are:
\[ \vec{E} = \sqrt{\frac{2I}{\epsilon_0 c}} \cos\left(\frac{2\pi}{\lambda}(y - ct)\right) \hat{k} \]
\[ \vec{B} = \frac{1}{c} \sqrt{\frac{2I}{\epsilon_0 c}} \cos\left(\frac{2\pi}{\lambda}(y - ct)\right) \hat{i} \]