Question

Consider each of the following electric- and magnetic-field orientations. In each case, what is the direction of propagation of the wave? (a)$\overrightarrow{\mathbf{E}}\mathbf{=}\mathbf{E}\stackrel{\mathbf{⏜}}{\mathbf{i}}\mathbf{,}\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{-}\mathbf{B}\stackrel{\mathbf{⏜}}{\mathbf{j}}\mathbf{;}$ (b)$\overrightarrow{\mathbf{E}}\mathbf{=}\mathbf{E}\stackrel{\mathbf{⏜}}{\mathbf{j}}\mathbf{,}\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{B}\stackrel{\mathbf{⏜}}{\mathbf{i}}\mathbf{;}$ (c)$\overrightarrow{\mathbf{E}}\mathbf{=}\mathbf{-}\mathbf{E}\stackrel{\mathbf{⏜}}{\mathbf{k}}\mathbf{,}\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{-}\mathbf{B}\stackrel{\mathbf{⏜}}{\mathbf{i}\mathbf{;}}$ (d) $\overrightarrow{\mathbf{E}}\mathbf{=}\mathbf{E}\stackrel{\mathbf{⏜}}{\mathbf{i}}\mathbf{,}\mathbf{}\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{-}\mathbf{B}\stackrel{\mathbf{⏜}}{\mathbf{k}}$

Short answer

a) The direction of propagation of the wave$\overrightarrow{E}=E\stackrel{⏜}{i}$ and$\overrightarrow{B}=-B\stackrel{⏜}{j}$ is -z- direction.

b) The direction of propagation of the wave$\overrightarrow{E}=E\stackrel{⏜}{j}$ and$\overrightarrow{B}=B\stackrel{⏜}{i}$ is -z- direction.

c) The direction of propagation of the wave$\overrightarrow{E}=-E\stackrel{⏜}{k}$ androle="math" localid="1664252484342" $\overrightarrow{B}=-B\stackrel{⏜}{i}$ is +y- direction.

d) The direction of propagation of the wave$\overrightarrow{E}=E\stackrel{⏜}{i}$ and$\overrightarrow{B}=-B\stackrel{⏜}{k}$ is -y- direction.

Step-by-step solution

Step 1: Define direction waves.

The electric and magnetic fields are perpendicular to the direction of propagation and to each other. The direction of the propagation is the given by the direction of $\overrightarrow{\mathbf{E}}\mathbf{\times }\overrightarrow{\mathbf{B}}$.

The vector equation of electromagnetic wave is:

$\overrightarrow{S}=\frac{1}{{\mu }_0}\left(\overrightarrow{E}\times \overrightarrow{B}\right)$

Determine the direction of wave.

The direction wave determined by applying right hand rule.

a) When the direction of electric field is positive x and the direction of magnetic is negative y, the direction of wave of propagation will be the negativez to become perpendicular to the electric and magnetic field.

$\stackrel{⏜}{i}\times \left(-\stackrel{⏜}{j}\right)=-\stackrel{⏜}{k}$

b) When the direction of electric field is positivey and the direction of magnetic is positive x, the direction of wave of propagation will be the negativez to become perpendicular to the electric and magnetic field.

$\stackrel{⏜}{j}\times \stackrel{⏜}{i}=-\stackrel{⏜}{k}$

c) When the direction of electric field is negative zand the direction of magnetic is negative x, the direction of wave of propagation will be the positivey to become perpendicular to the electric and magnetic field.

$\left(-\stackrel{⏜}{k}\right)\times \left(-\stackrel{⏜}{i}\right)=\stackrel{⏜}{j}$

d) When the direction of electric field is positive x and the direction of magnetic is negative z, the direction of wave of propagation will be the positive y to become perpendicular to the electric and magnetic field.$\stackrel{⏜}{i}\times \left(-\stackrel{⏜}{k}\right)=\stackrel{⏜}{j}$

Hence, direction of propagation of the wave $\overrightarrow{E}=E\stackrel{⏜}{i}$and $\overrightarrow{B}=-B\stackrel{⏜}{j}$is -z- direction, the direction of propagation of the wave $\overrightarrow{E}=E\stackrel{⏜}{j}$and $\overrightarrow{B}=B\stackrel{⏜}{i}$is -z- direction, the direction of propagation of the wave $\overrightarrow{E}=-E\stackrel{⏜}{k}$and $\overrightarrow{B}=-B\stackrel{⏜}{i}$is +y- direction, and the direction of propagation of the wavelocalid="1664253014464" $\overrightarrow{E}=E\stackrel{⏜}{i}$and localid="1664253026379" $\overrightarrow{B}=-B\stackrel{⏜}{k}$is +y- direction.