Question

An electromagnetic wave travelling along $z$-axis is given as $E=E_{0}cos(kz-\omega t)$. Choose the correct options from the following :

• A. The associated magnetic field is given as
  $\vec{B}=\frac{1}{c}\hat{k}\times \vec{E}=\frac{1}{\omega }\left(\vec{k}\times \vec{E}\right)$
• B. The electromagnetic field can be written in terms of the associated magnetic field as $\vec{E}=c\left(\vec{B}\times \hat{j}\right)$.
• C. $\hat{k}\cdot \vec{E}=0$, $\hat{k}\cdot \vec{B}\ne 0$
• D. $\hat{k}\cdot \vec{E}=0$, $\hat{k}\times \vec{B}=0$

Answer 1

Answer: (A)

We know that, direction of propagation of electromagnetic wave is always along the direction of vector product $\left(\vec{E}\times \vec{B}\right)$
$\vec{B}=B\hat{j}=B\left(\hat{k}\times \hat{i}\right)=\frac{E}{c}\left(\hat{k}\times \hat{i}\right)$
$\Rightarrow \vec{B}=\frac{1}{c}\left[\hat{k}\times E\hat{i}\right]=\frac{1}{c}\left[\hat{k}\times \vec{E}\right]$
If we write the equation in terms of magnetic field, then
$\vec{E}=c\left(\vec{B}\times \vec{k}\right)$.
$\hat{k}\cdot \vec{E}=\hat{k}\cdot \left(E\hat{i}\right)=0$,
$\hat{k}\cdot \hat{B}=\hat{k}\cdot \left(B\hat{j}\right)=0$,
as angle between them is $90^{\circ }$.