Question

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?

• A. $\hat{j}+\hat{k}, \hat{j}+\hat{k}$
• B. $-\hat{j}+\hat{k},-\hat{j}-\hat{k}$
• C. $\hat{j}+\hat{k},-\hat{j}-\hat{k}$
• D. $-\hat{j}+\hat{k},-\hat{j}+\hat{k}$

Answer 1

The Correct Option is B

To determine the correct possible directions for the electric field \((\mathbf{E})\) and magnetic field \((\mathbf{B})\) in a plane electromagnetic wave propagating in the \(x\)-direction, we need to apply the right-hand rule from electromagnetism. According to this principle:

1. The direction of propagation of an electromagnetic wave is determined by the cross product of the electric field \((\mathbf{E})\) and the magnetic field \((\mathbf{B})\). Mathematically, this is described as \(\mathbf{E} \times \mathbf{B} = \text{direction of propagation}\).
2. Since the wave propagates in the \(x\)-direction, the cross product \(\mathbf{E} \times \mathbf{B}\) must also point in the \(x\)-direction.
3. Given this condition, if we choose the directions of \(\mathbf{E}\) and \(\mathbf{B}\) as vectors, they need to be perpendicular to the direction of propagation and to each other.
4. The correct option is \(-\hat{j}+\hat{k}, -\hat{j}-\hat{k}\) because:
   • If \(\mathbf{E} = -\hat{j} + \hat{k}\), it lies in the \(y-z\) plane, forming a perpendicular vector to the direction of propagation \(x\).
   • If \(\mathbf{B} = -\hat{j} - \hat{k}\), it is also in the \(y-z\) plane but forms a suitable pair such that \(\mathbf{E} \times \mathbf{B}\) results in \(+ \hat{i}\) (i.e., positive \(x\)-direction).

Thus, among the given options, \(-\hat{j}+\hat{k}, -\hat{j}-\hat{k}\) is the correct one based on the right-hand rule and vector analysis in electromagnetism.