A plane electromagnetic wave of wavelength $$\lambda$$ has an intensity I. It is propagating along the positive Y-direction. The allowed expressions for the electric and magnetic fields are given by:

For a plane electromagnetic wave, the direction of energy transport is given by the Poynting vector $$\vec S = \frac{1}{\mu_0}\,\vec E \times \vec B$$. If the wave is travelling along the positive Y-axis, then $$\vec E$$ and $$\vec B$$ must both be perpendicular to the Y-axis and also perpendicular to each other in such a way that $$\vec E \times \vec B$$ points along $$+\hat j$$.

A convenient perpendicular pair is to take the electric field along the Z-axis and the magnetic field along the X-axis, because then

$$\hat k \times \hat i = \hat j,$$

which indeed points in the positive Y-direction. So we put

$$\vec E = E_0 \cos\!\bigl(ky - \omega t\bigr)\,\hat k, \qquad \vec B = B_0 \cos\!\bigl(ky - \omega t\bigr)\,\hat i.$$

Next we recall two standard relations for electromagnetic waves in vacuum:

1. The magnitudes are related by $$B_0 = \dfrac{E_0}{c}.$$

2. The intensity $$I$$ (time-averaged magnitude of the Poynting vector) is

$$I = \frac{1}{2}\,\varepsilon_0 c\,E_0^{\,2}.$$

From the intensity formula we can solve for $$E_0$$ step by step:

$$I = \frac{1}{2}\,\varepsilon_0 c\,E_0^{\,2} \quad\Longrightarrow\quad E_0^{\,2} = \frac{2I}{\varepsilon_0 c} \quad\Longrightarrow\quad E_0 = \sqrt{\frac{2I}{\varepsilon_0 c}}.$$

Substituting $$E_0$$ into the expression for $$B_0$$ we get

$$B_0 = \frac{E_0}{c} = \frac{1}{c}\sqrt{\frac{2I}{\varepsilon_0 c}}.$$

The phase factor for a wave travelling in the +Y direction must be $$\bigl(ky - \omega t\bigr)$$, or equivalently $$\frac{2\pi}{\lambda}(y - ct)$$, because $$k = \frac{2\pi}{\lambda}$$ and $$\omega = ck$$. Using any other sign, such as $$y + ct$$, would represent a wave travelling in the negative Y-direction, which is not the situation given in the problem.

Putting everything together we obtain the allowed field expressions:

$$\vec E = \sqrt{\frac{2I}{\varepsilon_0 c}}\, \cos\!\left[\frac{2\pi}{\lambda}(y - ct)\right]\hat k,$$

$$\vec B = \frac{1}{c}\,E\;\hat i.$$

Comparing this result with the options, we see that it coincides exactly with Option A, while the other options either use the wrong amplitude, the wrong phase sign, or swap the field directions.

Hence, the correct answer is Option A.