Question 25: EM waves

Question

For an electromagnetic wave propagating through vacuum, $\vec{k}, \vec{E}$ and $\omega$ represent propagation vector, electric field and angular frequency, respectively. The magnetic field associated with this wave is represented by:

Accepted Answer

In an electromagnetic wave, the electric field $\vec{E}$, magnetic field $\vec{B}$, and the propagation direction $\vec{k}$ are mutually perpendicular to each other. The wave travels in the direction of the cross product $\overrightarrow{\mathrm{E}} 	imes \overrightarrow{\mathrm{B}}$.
Also the magnetic field vector is in the direction of $\widehat{\mathrm{k}} 	imes \widehat{\mathrm{E}}$
The magnitude of magnetic field is given as $\mathrm{B}=\frac{\mathrm{E}}{\mathrm{c}}$, where c is the speed of light.
Maxwell's equation (Faraday's Law) in differential form is :
$$ 
abla 	imes \overrightarrow{\mathrm{E}}=-\frac{\partial \overrightarrow{\mathrm{B}}}{\partial \mathrm{t}} $$
For a plane monochromatic wave, the fields can be represented asb :
$$ \overrightarrow{\mathrm{E}}=\overrightarrow{\mathrm{E}}_0 \mathrm{e}^{\mathrm{i}(\overrightarrow{\mathrm{k}} \cdot \overrightarrow{\mathrm{r}}-\omega \mathrm{t})} $$
$$ \overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{B}}_0 \mathrm{e}^{\mathrm{i}(\overrightarrow{\mathrm{k}} \cdot \overrightarrow{\mathrm{r}}-\omega \mathrm{t})} $$
Applying the curl operator ( $
abla 	imes$ ) to the electric field expression :
$$ 
abla 	imes \overrightarrow{\mathrm{E}}=\mathrm{i} \overrightarrow{\mathrm{k}} 	imes \overrightarrow{\mathrm{E}} $$
Applying the time derivative to the magnetic field expression :
$$ \frac{\partial \overrightarrow{\mathrm{B}}}{\partial \mathrm{t}}=-\mathrm{i} \omega \overrightarrow{\mathrm{~B}} $$
Substituting into Faraday's Law :
$$ \mathrm{i} \overrightarrow{\mathrm{k}} 	imes \overrightarrow{\mathrm{E}}=-(-\mathrm{i} \omega \overrightarrow{\mathrm{~B}}) $$
$$\Rightarrow $$ $\mathrm{i} \overrightarrow{\mathrm{k}} 	imes \overrightarrow{\mathrm{E}}=\mathrm{i} \omega \overrightarrow{\mathrm{B}}$
$$\Rightarrow $$ $$ \overrightarrow{\mathrm{B}}=\frac{\overrightarrow{\mathrm{k}} 	imes \overrightarrow{\mathrm{E}}}{\omega} $$
Therefore, the correct option is (B)

Answer

$\frac{\vec{E} 	imes \vec{k}}{\omega}$

Answer

$\frac{\vec{k} 	imes \vec{E}}{\omega}$

Answer

$\omega(\vec{E} 	imes \vec{k})$

Answer

$\omega(\vec{k} 	imes \vec{E})$

EM waves

For an electromagnetic wave propagating through vacuum, \vec{k}, \vec{E} and \omega represent propagation vector, electric field and angular frequency, respectively. The magnetic field associated with this wave is represented by:

Choose the correct answer: