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 $\left(\vec{E} \times \vec{B}\right)$
$\vec{B} = B\hat{j} = B\left(\hat{k} \times \hat{i}\right) = \frac{E}{c}\left(\hat{k} \times \hat{i}\right)$
$\Rightarrow \quad \vec{B} = \frac{1}{c} \left[\hat{k} \times E \hat{i}\right] = \frac{1}{c} \left[\hat{k} \times \vec{E}\right]$
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 $90^{\circ}$.