A plane electromagnetic wave of frequency 20 MHz propagates in free space along $$x$$-direction. At a particular space and time $$\vec{E} = 6.6\hat{j}$$ V m$$^{-1}$$. What is $$\vec{B}$$ at this point?

The wave propagates along the $$x$$-direction with $$\vec{E} = 6.6\hat{j}$$ V/m.

For an EM wave, $$\vec{E} \times \vec{B}$$ must be along the direction of propagation. Since propagation is along $$\hat{i}$$ and $$\vec{E}$$ is along $$\hat{j}$$:

$$\hat{j} \times \hat{k} = \hat{i}$$ ✓

So $$\vec{B}$$ is along $$\hat{k}$$.

Magnitude: $$B = \frac{E}{c} = \frac{6.6}{3 \times 10^8} = 2.2 \times 10^{-8}$$ T

$$\vec{B} = 2.2 \times 10^{-8}\hat{k}$$ T