A plane electromagnetic wave of frequency 100 MHz is traveling in a vacuum along the $$x$$-direction. At a particular point in space and time, $$\vec{B} = 2.0 \times 10^{-8}\hat{k}$$ T. (where, $$\hat{k}$$ is unit vector along $$z$$-direction) What is $$\vec{E}$$ at this point?

For an electromagnetic wave traveling in vacuum along the $$x$$-direction, the electric and magnetic fields are perpendicular to each other and to the direction of propagation. Given $$\vec{B} = 2.0 \times 10^{-8}\hat{k}$$ T, we need $$\vec{E}$$ such that the Poynting vector $$\vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}$$ points along $$\hat{i}$$.

If $$\vec{E}$$ is along $$\hat{j}$$, then $$\hat{j} \times \hat{k} = \hat{i}$$, which correctly gives the propagation direction. The magnitude is $$E = cB = (3 \times 10^8)(2.0 \times 10^{-8}) = 6.0$$ V/m.

Therefore $$\vec{E} = 6.0\hat{j}$$ V m$$^{-1}$$.