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((I)/(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((2I)/(in_(0)c))cos[(2pi)/(lambda)(y-ct)]hati`,
D. `vecB=(1)/(c)E " veci`

Answer

To solve the problem, we need to analyze the given information about the plane electromagnetic wave and derive the expressions for the electric and magnetic fields based on the intensity \( I \) and wavelength \( \lambda \).

### Step-by-Step Solution:

1. **Understanding the Direction of Propagation**:
The electromagnetic wave is propagating along the positive Y-direction. This means the wave vector \( \vec{k} \) is in the direction of \( \hat{j} \) (the unit vector in the Y-direction).

2. **Identifying the Electric and Magnetic Fields**:
The electric field \( \vec{E} \) and magnetic field \( \vec{B} \) are perpendicular to each other and also perpendicular to the direction of wave propagation. For a wave propagating in the Y-direction:
- The electric field \( \vec{E} \) can be in the Z-direction (along \( \hat{k} \)).
- The magnetic field \( \vec{B} \) can be in the X-direction (along \( \hat{i} \)).

3. **Using the Intensity Formula**:
The intensity \( I \) of an electromagnetic wave is related to the electric field \( E_0 \) and the speed of light \( c \) by the formula:
\[ I = \frac{1}{2} \epsilon_0 c E_0^2 \]
where \( \epsilon_0 \) is the permittivity of free space.

4. **Solving for \( E_0 \)**:
Rearranging the intensity formula to solve for \( E_0 \):
\[ E_0 = \sqrt{\frac{2I}{\epsilon_0 c}} \]

5. **Constructing the Electric Field Expression**:
The general expression for the electric field of a plane wave can be written as:
\[ \vec{E} = E_0 \cos(k y - \omega t) \hat{k} \]
where \( k = \frac{2\pi}{\lambda} \) is the wave number and \( \omega = 2\pi f \) is the angular frequency.

6. **Substituting for \( E_0 \)**:
Substitute \( E_0 \) into the electric field expression:
\[ \vec{E} = \sqrt{\frac{2I}{\epsilon_0 c}} \cos\left(\frac{2\pi}{\lambda}(y - ct)\right) \hat{k} \]

7. **Final Expression for the Electric Field**:
The final expression for the electric field is:
\[ \vec{E} = \sqrt{\frac{2I}{\epsilon_0 c}} \cos\left(\frac{2\pi}{\lambda}(y - ct)\right) \hat{k} \]