Question

An electromagnetic wave is traveling in the negative direction of ay-axis. At a particular position and time, the electric field is directed along the positive direction of thez-axis and has a magnitude of $\mathbf{100}\mathbf{}\mathbf{\text{V/m}}$. What are the (a) magnitude and (b) direction of the corresponding magnetic field?

Short answer

a. The magnitude of the magnetic field is $3.3\times {10}^{-7}\text{T}$.

b. The direction of the magnetic field is along the negative x-axis.

Step-by-step solution

Given Information

The direction of an electromagnetic wave is along the negative y-axis.

The direction of the electric field is along the positive z-axis.

The magnitude of the electric field is, $\left|\overrightarrow{\text{E}}\right|=100\text{V/m}$.

Understanding the concept of wave propagation

Here, we need to use the equation for the magnetic field associated with an electromagnetic wave to find the magnitude of the magnetic field. The direction of the magnetic field can be calculated using the vector identity of the cross product that the propagating wave is perpendicular to both the electric and magnetic field directions.

Formulae:

The magnitude of the magnetic field of a light wave,

$\text{B}=\frac{\text{E}}{\text{c}}$...........................(1)

The vector identity of three directional vectors, $∆k\times ∆i=∆j$.............................(2)

a) Calculation of the magnitude of the magnetic field

Using the given data in equation (1), we can get the magnitude of the magnetic field as follows:

$\begin{array}{rcl}B& =& \frac{100\text{V/m}}{3\times {10}^8\text{m/s}}\\ & =& 3.3\times {10}^{-7}\text{T}\\ & & \end{array}$

Hence, the value of the magnetic field is $3.3\times {10}^{-7}\text{T}$.

b) Calculation of the direction of the magnetic field 

As the electromagnetic wave is along the negative y-axis $\left(-\hat{j}\right)$, the direction of the magnetic field also must be along the negative axis, that is,$\left(-\hat{i}\right)$ according to equation (2), as the direction of the electric field is along the positive$\left(+k\right)$ axis.

Hence, the direction of the magnetic field is along the negative x-axis.