Question

An electromagnetic wave travelling along z-axis is given as E = $E_0\cos (kz-\omega t)$. Choose the correct option from the following :

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

Answer 1

The correct option is C $k\cdot \overrightarrow{E}=0,k\cdot \overrightarrow{B}\ne 0.$

Since the direction of propagation of the wave is along the Z-axis the Electric field will be perpendicular to the Z-axis will be acting in the negative Y-axis.

Then the associated magnetic field will be along the direction $(\hat{k}\times \hat{E})$

$\overrightarrow{B}=\frac{1}{c}(\hat{k}\times \hat{E})=\frac{1}{\omega }(\hat{k}\times \hat{E})$

Now, the electric field will be along the direction $(\overrightarrow{B}\times \hat{k})$ i.e perpendicular to the magnetic field and direction of motion.

The electromagnetic field can be written in terms of the associated magnetic field as

$\overrightarrow{E}=c(\overrightarrow{B}\times \hat{k})$

The angle between the $\hat{k}$ and $\overrightarrow{E}$ will be ${90}^0$ whereas between the $\hat{k}$ and $\overrightarrow{B}$ will be ${180}^0$

So, here, $\hat{k}.\overrightarrow{E}=0,\hat{k}.\overrightarrow{B}\ne 0$
Hence, the correct option is $\left(C\right)$