Electromagnetic Waves Part 2

Concept:In an electromagnetic wave, the electric field ($\mathbf{E}$) and magnetic field ($\mathbf{B}$) are perpendicular to each other and to the direction of propagation.Their amplitudes are related by $E_0 = c B_0$, where $c$ is the speed of light.Explanation:The given electric field is $E_y = 69 \sin(0.6\times10^3 x - 1.8\times10^{11}t)$ V/m.The amplitude is $E_0 = 69$ V/m.The speed of light is $c = 3\times10^8$ m/s.Thus $B_0 = \frac{E_0}{c} = \frac{69}{3\times10^8} = 2.3\times10^{-7}$ T.The wave propagates along the $+x$ direction because the phase term is $(kx - \omega t)$ with positive $k$ and $\omega$.Electric field oscillates along the $y$-axis.Using the right-hand rule: $\hat{s} = \hat{E} \times \hat{B}$, where $\hat{s}$ is propagation direction.Here $\hat{s} = \hat{i}$, $\hat{E} = \hat{j}$, so $\hat{i} = \hat{j} \times \hat{B}$ implies $\hat{B} = \hat{k}$ (magnetic field along $z$).Electric and magnetic fields are in phase in a vacuum.Therefore, the magnetic field vector is $B_z = 2.3\times10^{-7} \sin(0.6\times10^3 x - 1.8\times10^{11}t)$ T.This matches option A.Answer:Option A: $B_z = 2.3 \times 10^{-7} \sin [0.6 \times 10^3 x - 1.8 \times 10^{11} t]$ T.