Question: 17. For a plane electromagnetic wave propagating in x-direction, which one of the following combination gives the correct possible directions for electric field (E) and magnetic field (B) respectively ?
(1) \hat{j}+\hat{k}, \hat{j}+\hat{k}
(2) -\hat{j}+\hat{k},-\hat{j}-\hat{k}
(3) \hat{j}+\hat{k},-\hat{j}-\hat{k}
(4) -\hat{j}+\hat{k},-\hat{j}+\hat{k}
Answer: Option (2)
Explanation:
In a plane electromagnetic wave, the electric field \vec{E},
magnetic field \vec{B}, and direction of propagation are mutually perpendicular.
The direction of propagation of the wave is given by the vector product \vec{E}\times\vec{B}.
For propagation along the positive x-direction,
\vec{E}\times\vec{B} must be along \hat{i}.
Also, the electric and magnetic fields must be perpendicular to each other,
so \vec{E}\cdot\vec{B}=0.
For option (2), the directions are \vec{E}=-\hat{j}+\hat{k} and \vec{B}=-\hat{j}-\hat{k}.
Their dot product is (-\hat{j}+\hat{k})\cdot(-\hat{j}-\hat{k})=1-1=0,
so the fields are perpendicular.
Their cross product is (-\hat{j}+\hat{k})\times(-\hat{j}-\hat{k})=2\hat{i},
which is along the positive x-direction.
Hence, this combination represents a plane electromagnetic wave propagating along the x-direction.
Therefore, the correct answer is Option (2).