Question

For plane electromagnetic waves propagating in the z direction, which one of the following combination gives the correct possible direction for $\overrightarrow{E}$ and $\overrightarrow{B}$ field respectively?

• A. $(\hat{i}+2\hat{j})$ and $(2\hat{i}-\hat{j})$
• B. $(-2\hat{i}-3\hat{j})$ and $(3\hat{i}-2\hat{j})$
• C. $(2\hat{i}+3\hat{j})$ and $(\hat{i}+2\hat{j})$
• D. $(3\hat{i}+4\hat{j})$ and $(4\hat{i}-3\hat{j})$

Answer 1

Answer: (B)

$(-2\hat{i}-3\hat{j})$ and $(3\hat{i}-2\hat{j})$

The $\overrightarrow{E}$ and $\overrightarrow{B}$ are $\perp$ to each other, their dot product must be zero
$\overrightarrow{E}.\overrightarrow{B}=0$
Checking option A $(\hat{i}+2\hat{j}).(2\hat{i}-\hat{j})=0$

Checking option B $(-2\hat{i}-3\hat{j}).(3\hat{i}-2\hat{j})=0$

Checking option C $(2\hat{i}+3\hat{j}).(\hat{i}+2\hat{j})=8$ it can not be the answer

Checking option C $(3\hat{i}+4\hat{j}).(4\hat{i}-3\hat{j})=0$

For wave to travel in positive z direction it's direction should be of form $\hat{k}$, the direction of wave can be determined using cross product of E and B:

Checking option A $(\hat{i}+2\hat{j})\times (2\hat{i}-\hat{j})=-\hat{k}-4\hat{k}=-5\hat{k}$ it can not be the answer

Checking option B $(-2\hat{i}-3\hat{j})\times (3\hat{i}-2\hat{j})=4\hat{k}+9\hat{k}=13\hat{k}$

Checking option D $(3\hat{i}+4\hat{j})\times (4\hat{i}-3\hat{j})=-12\hat{k}-12\hat{k}=-24\hat{k}$ it can not be the answer.

Hence, option B is the answer.