Question

For plane electromagnetic waves which are propagating in the z direction, which one of the following combinations gives the correct possible direction for the E and B fields respectively?
$\begin{align}
  & A.\left( \widehat{i}+2\widehat{j} \right)and\left( 2\widehat{i}-\widehat{j} \right) \\
 & B.\left( -2\widehat{i}-3\widehat{j} \right)and\left( 3\widehat{i}-2\widehat{j} \right) \\
 & C.\left( 2\widehat{i}+3\widehat{j} \right)and\left( \widehat{i}+2\widehat{j} \right) \\
 & D.\left( 3\widehat{i}+4\widehat{j} \right)and\left( 4\widehat{i}-3\widehat{j} \right) \\
\end{align}$

Answer 1

Hint: If we thoroughly check the question, we will get to know that, $\overrightarrow{E}and\overrightarrow{B}$ are perpendicular to each other. Therefore their dot product will be zero. This will help us to solve the equation. And the direction of motion of the electromagnetic wave will be the cross product of $\overrightarrow{E}and\overrightarrow{B}$.

Complete step by step answer:
In this question as$\overrightarrow{E}and\overrightarrow{B}$are perpendicular to each other, their dot product will be zero.
$\overrightarrow{E}\centerdot \overrightarrow{B}=0$
\[\left( x\widehat{i}+y\widehat{j} \right)\centerdot \left( a\widehat{i}+b\widehat{j} \right)=\left( xa+yb \right)\]\[\overrightarrow{E}\times \overrightarrow{B}=z\widehat{k}\]
For finding the answer let us check each and every option regarding these relations.

First option,
$(\widehat{i}+2\widehat{j}).2\widehat{i}-\widehat{j})=0$$\left( 3\times 4 \right)+\left( 4\times -3 \right)=0$
$\left( 1\times 2 \right)-\left( 2\times -1 \right)=0$
Second option will give,
$(-2\widehat{i}-3\widehat{j}).(3\widehat{i}-2\widehat{j})=0$
$\left( -2\times 3 \right)+\left( -3\times -2 \right)=0$
Now let us check the third option,
$(2\widehat{i}+3\widehat{j}).(\widehat{i}+2\widehat{j})=8$
$\left( 2\times 1 \right)+\left( 3\times 2 \right)=8$
The dot product is not giving zero. Therefore option C will not be the answer.
Now let us check option D
$(3\widehat{i}+4\widehat{j}).(4\widehat{i}-3\widehat{j})=0$
$\left( 3\times 4 \right)+\left( 4\times -3 \right)=0$
Therefore only three options are giving the dot product as zero.
Now let us check the direction of the electromagnetic waves. We can see that the waves travelling in the z direction, which is perpendicular to$\overrightarrow{E}and\overrightarrow{B}$.
\[\overrightarrow{E}\times \overrightarrow{B}=z\widehat{k}\]
\[\left( x\widehat{i}+y\widehat{j} \right)\times \left( a\widehat{i}+b\widehat{j} \right)=\left( yz\widehat{i}+xz\widehat{j}+xy\widehat{k} \right)\]
Therefore the direction can be found out using the cross product which should be positive.
Now let us check the first option A
$(\widehat{i}+2\widehat{j})\times (2\widehat{i}-\widehat{j})=-\widehat{k}-4\widehat{k}=-5\widehat{k}$
$\left| \begin{align}
  & \text{i j k} \\
 & \text{1 2 0} \\
 & \text{2 -1 0} \\
\end{align} \right|=-5\widehat{k}$
This is negative, so this will not be the correct answer.
Let us check option B
$(-2\widehat{i}-3\widehat{j})\times (3\widehat{i}-2\widehat{j})=4\widehat{k}+9\widehat{k}=13\widehat{k}$
$\left| \begin{align}
  & \text{i j k} \\
 & \text{-2 -3 0} \\
 & \text{3 -2 0} \\
\end{align} \right|=13\widehat{k}$
This may be a possible choice.
Let us check option C,
$\left| \begin{align}
  & \text{i j k} \\
 & \text{2 3 0} \\
 & \text{1 2 0} \\
\end{align} \right|=\left( 4-3 \right)\widehat{k}=\widehat{k}$
This is also negative, therefore will not be the correct answer.
Anyway let us check the option D also.
\[(3\widehat{i}+4\widehat{j})\times (4\widehat{i}-3\widehat{j})=-9\widehat{k}-16\widehat{k}=-25\widehat{k}\]
$\left| \begin{align}
  & \text{i j k} \\
 & \text{3 4 0} \\
 & \text{4 -3 0} \\
\end{align} \right|=-25\widehat{k}$
This will not be the answer as it is negative.
Hence the correct answer is option B.

Note:
Electromagnetic waves are the waves that are produced as a result of vibrations in between an electric field and a magnetic field. Another way, electromagnetic waves are formed up of oscillating magnetic and electric fields.