Question

A sinusoidal electromagnetic wave is propagating in vacuum in the +z- direction. If at a particular instant and at a certain point in space the electric field is in the +x- direction and has magnitude what are the magnitude 4.00 V/m and direction of the magnetic field of the wave at this same point in space and instant in time?

Short answer

The magnitude of magnetic field is $1.33\times {10}^{-8}\mathrm{T}$ and the direction of the wave 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:

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

The formula used to determine the amplitude of electric and magnetic fields of the wave are:

$$\begin{gathered} E_{max}=c{B}_{max} \\ {B}_{max}=\frac{E_{max}}{c} \end{gathered}$$

Determine the direction and magnitude of wave.

The direction wave determined by applying right hand rule.

When the direction of electric field is positivex and the direction of magnetic is positive z, the direction of wave of propagation will be the positivey to become perpendicular to the electric and magnetic field.

The formula used to determine the amplitude of electric and magnetic fields of the wave are:

$$\begin{gathered} {\mathrm{B}}_{\max }=\frac{{\mathrm{E}}_{\max }}{\mathrm{c}} \\ =\frac{4.0}{3\times {10}^8} \\ =1.33\times {10}^{-8}\mathrm{T} \end{gathered}$$

Hence, magnitude of magnetic field is $1.33\times {10}^{-8}\mathrm{T}$and the direction of the wave is +y- direction.