Electromagnetic Waves Part 2

Concept:In an electromagnetic wave in free space, the electric field $\vec{E}$ and magnetic field $\vec{B}$ are perpendicular to each other and to the direction of propagation.Their amplitudes are related by $c = E_0 / B_0$, where $c$ is the speed of light.Explanation:Amplitude of electric field $E_0 = 54 \, \text{V/m}$, speed $c = 3 \times 10^8 \, \text{m/s}$.So $B_0 = E_0 / c = 54 / (3 \times 10^8) = 1.8 \times 10^{-7} \, \text{T}$.The wave propagates along $+z$ direction (given by $kz - \omega t$), so unit vector along propagation is $\hat{k}$.Electric field is along $\hat{j}$ (y-axis).Using right-hand rule: $\hat{E} \times \hat{B} = \hat{v}$.We need $\hat{j} \times \hat{B} = \hat{k}$.Since $\hat{j} \times \hat{i} = \hat{k}$? Actually, $\hat{j} \times \hat{i} = -\hat{k}$. To get $+\hat{k}$, we need $\hat{j} \times (-\hat{i}) = \hat{k}$.Thus $\hat{B} = -\hat{i}$, i.e., magnetic field is along negative x-direction.Combine magnitude and direction: $\vec{B} = -1.8 \times 10^{-7} \sin(kz - \omega t) \, \hat{i} \, \text{T}$.Answer:Option A: $-1.8 \times 10^{-7} \sin(kz - \omega t) \hat{i} \, \text{T}$.