Question

Consider each of the electric- and magnetic-field orientations given next. In each case, what is the direction of propagation of the wave? (a) $\overrightarrow{\mathbf{E}}$in the +x- direction, $\overrightarrow{\mathbf{B}}$in the +y- direction; (b)$\overrightarrow{\mathbf{E}}$ in the -y- direction,$\overrightarrow{\mathbf{B}}$ in the +x- direction; (c)$\overrightarrow{\mathbf{E}}$ in the +z- direction,$\overrightarrow{\mathbf{B}}$ in the -x- direction; (d)$\overrightarrow{\mathbf{E}}$ in the +y- direction,$\overrightarrow{\mathbf{B}}$ in the +z- direction.

Short answer

a) The direction of propagation of the wave $\overrightarrow{E}$in +x- direction,$\overrightarrow{B}$in +y- direction is +z- direction.

b) The direction of propagation of the wave$\overrightarrow{E}$ in -y- direction,$\overrightarrow{B}$ in +x- direction is +z- direction.

c) The direction of propagation of the wave $\overrightarrow{E}$in +z- direction, $\overrightarrow{B}$in +x- direction is +z- direction.

d) The direction of propagation of the wave $\overrightarrow{E}$in +y- direction,$\overrightarrow{B}$ in -z- direction is -x- 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 positivex and the direction of magnetic is positive y, the direction of wave of propagation will be the positivez to become perpendicular to the electric and magnetic field.

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

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

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

Hence, the direction of propagation of the wave $\overrightarrow{E}$in +x- direction,localid="1664250994986" $\overrightarrow{B}$in +y- direction is +z- direction, the direction of propagation of the wave$\overrightarrow{E}$ in -Y- direction,$\overrightarrow{B}$ in +x-direction is +z-direction, the direction of propagation of the wavelocalid="1664250998038" $\overrightarrow{E}$in +z- direction,$\overrightarrow{B}$in +x- direction is -y- direction and the direction of propagation of the wave $\overrightarrow{E}$in +y- direction, $\overrightarrow{B}$in -z- direction is -x- direction.