For plane electromagnetic waves propagating in the $$+z$$-direction, which one of the following combinations gives the correct possible direction for $$\vec{E}$$ and $$\vec{B}$$ field respectively?

We are told that a plane electromagnetic (EM) wave is propagating in the $$+z$$-direction. For such a wave we recall two basic facts from Maxwell’s theory:

1. The electric field $$\vec E$$, the magnetic field $$\vec B$$ and the propagation vector $$\vec k$$ are mutually perpendicular. Since the wave travels along $$+z$$, we have $$\vec k = k\,\hat z$$, so both $$\vec E$$ and $$\vec B$$ must lie completely in the $$xy$$-plane; they can have $$\hat i$$ and $$\hat j$$ components but no $$\hat k$$ component.

2. The quantitative relation between the three vectors is

$$\vec k \times \vec E = k\,\hat z \times \vec E = \dfrac{\omega}{c}\,\vec B,$$

or, after introducing the unit vector $$\hat k = \hat z$$, more compactly

$$\;\vec B = \dfrac1c\,\hat k \times \vec E\;.$$

Because the constant factor $$1/c$$ only rescales the vector, the important geometric statement is simply

$$\;\vec B \;||\; (\hat z \times \vec E)\;.$$

In words: “$$\vec B$$ must be the cross-product of $$\hat z$$ with $$\vec E$$, so its direction is fixed once $$\vec E$$ is chosen.” We will apply this to each option.

Write a general $$\vec E$$ in the $$xy$$-plane as

$$\vec E = E_x\,\hat i + E_y\,\hat j + 0\,\hat k.$$ Then

$$\hat z \times \vec E \;=\; \begin{vmatrix} \hat i & \hat j & \hat k\\ 0 & 0 & 1\\ E_x & E_y & 0 \end{vmatrix} =\hat i(0\cdot0-1\cdot E_y) -\hat j(0\cdot0-1\cdot E_x) + \hat k(0\cdot E_y-0\cdot E_x)$$

$$=\;(-E_y)\,\hat i + (E_x)\,\hat j + 0\,\hat k.$$ So the rule becomes

$$\;\vec B \propto (-E_y)\,\hat i + (E_x)\,\hat j\;. \cdots(*)$$

Now we test each given pair.

Option A: $$\vec E = (1)\hat i + (2)\hat j,\qquad \vec B = (2)\hat i + (-1)\hat j.$$ Using (*), the required $$\vec B$$ would be $$(-E_y)\hat i + (E_x)\hat j = (-2)\hat i + (1)\hat j,$$ which is the negative of the given $$\vec B$$. So Option A does not satisfy the rule.

Option B: $$\vec E = (-2)\hat i + (-3)\hat j,\qquad \vec B = (3)\hat i + (-2)\hat j.$$ Applying (*), $$(-E_y)\hat i + (E_x)\hat j = -(-3)\hat i + (-2)\hat j = 3\hat i - 2\hat j.$$ This is exactly the given $$\vec B$$. Hence Option B is consistent with electromagnetic-wave theory.

Option C: $$\vec E = (2)\hat i + (3)\hat j,\qquad \vec B = (1)\hat i + (2)\hat j.$$ Formula (*) demands $$(-E_y)\hat i + (E_x)\hat j = (-3)\hat i + (2)\hat j,$$ which differs from the given $$\vec B$$. Therefore Option C is invalid.

Option D: $$\vec E = (3)\hat i + (4)\hat j,\qquad \vec B = (4)\hat i + (-3)\hat j.$$ Rule (*) gives $$(-E_y)\hat i + (E_x)\hat j = (-4)\hat i + (3)\hat j,$$ again the negative of the listed $$\vec B$$, so Option D fails.

Only Option B satisfies the mandatory cross-product relation, so it is the only physically admissible choice.

Hence, the correct answer is Option B.