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

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

(a)    \hat{k}\times E=0,\hat{k}\times B=0

(b)    \hat{k}. E=0,\hat{k}\times B=0

(c)    the associated magnetic field is given as

B= \frac{1}{c}\hat{k} \times E=\frac{1}{\omega }\left ( \hat{k}\times E \right )

(d)    the electromagnetic field can be written in terms of the associated magnetic field as

E=c\left ( B\times \hat{k} \right )

Answers (1)

Answer: The correct answers are the options (b, c and d)

The direction of propagation of an electromagnetic wave is always along the direction of vector product \vec{E}\times \vec{B}. Refer to figure
(a)

\\\vec{B}=B\hat{j}=B\left ( \hat{k}\times \hat{i} \right )=\frac{E}{c}\left ( \hat{k}\times \hat{i} \right )\\\\ =\frac{1}{c}\left [ k\times E\hat{i} \right ]=\frac{1}{c}\left [ \hat{k} \times \vec{E} \right ]\: \: \: As\frac{E}{B}=c

 

(b) 
   \\\vec{E}=E\hat{i}=cB\left ( \hat{j} \times\hat{k}\right )=c\left (B \hat{j} \times\hat{k}\right )=c\left (\vec{B}\times \hat{k} \right )

(c)
    \\ \hat{k}\cdot \vec{E}=\hat{k}\cdot \left ( E\hat{i} \right )= 0,\vec{k}.\vec{B}=\hat{k}\cdot \left ( B\hat{j} \right )=0
(d)
\\ \hat{k}\times \vec{E}=\hat{k}\cdot \left ( E\hat{i} \right )= E\left ( \hat{k} \times \hat{i}\right )=E\hat{j}\\\\and\: \: \hat{k}\times \vec{B}= \hat{k}\times \left ( B\hat{j} \right )=B\left ( \hat{k}\times \hat{j} \right )=-B\hat{i}
 

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