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 solve this problem, we need to understand how electromagnetic waves propagate. For a plane electromagnetic wave traveling in the \(x\)-direction, the directions of the electric field (\(\mathbf{E}\)) and magnetic field (\(\mathbf{B}\)) should be perpendicular to each other and also perpendicular to the direction of propagation. This is described by the right-hand rule, as well as being consistent with Maxwell's equations.

1. The wave is propagating in the \(x\)-direction. Therefore, the \(\mathbf{E}\) and \(\mathbf{B}\) fields must lay in the \(yz\)-plane (perpendicular to \(x\)).
2. The options for the directions of \(\mathbf{E}\) and \(\mathbf{B}\) are given:
   • \(\hat{j}+\hat{k}, \hat{j}+\hat{k}\)
   • \(-\hat{j}+\hat{k},-\hat{j}-\hat{k}\)
   • \(\hat{j}+\hat{k},-\hat{j}-\hat{k}\)
   • \(-\hat{j}+\hat{k},-\hat{j}+\hat{k}\)
3. We apply the right-hand rule: if you point the thumb of your right hand in the direction of the wave propagation (i.e., \(x\)-direction), and your fingers in the direction of the electric field (\(\mathbf{E}\)), your palm points in the direction of the magnetic field (\(\mathbf{B}\)), or vice versa.
4. Analyzing the directions:
   • \(-\hat{j}+\hat{k}\) for \(\mathbf{E}\) implies the field is in the negative \(y\)-direction and positive \(z\)-direction.
   • \(-\hat{j}-\hat{k}\) for \(\mathbf{B}\) implies the field is in the negative \(y\)-direction and negative \(z\)-direction.
5. This orientation aligns with the perpendicular requirement and satisfies Maxwell’s equation constraints based on the given direction of wave propagation.
6. Consequently, the correct option is \(-\hat{j}+\hat{k},-\hat{j}-\hat{k}\), where both fields are perpendicular to the direction of wave propagation and each other. This also means that the curl of \(\mathbf{E}\) aligns properly with \(\mathbf{B}\), confirming this is a valid electromagnetic wave arrangement traveling in the x-direction.