Question

Which of the following pairs of the component of space and time varying $\mathrm{E}^{-}=\left(\mathrm{E}_{\mathrm{x}} \mathrm{i} \wedge+\mathrm{Eyj} \wedge+\mathrm{Ezk} \wedge\right)$ and $\mathrm{B}^{-}=\left(\mathrm{B}_{\mathrm{x}} \mathrm{i}^{\mathrm{i}}+\mathrm{Byj}^{\wedge}+\mathrm{Bzk} \wedge\right)$ would generate a plane electromagnetic wave travelling in \(+\) ve \(z\) direction (A) \(E x, B y\) (B) \(\mathrm{Ey}, \mathrm{Bz}\) (C) \(\mathrm{Ex}, \mathrm{Bz}\) (D) \(E z, B x\)

Short answer

The correct pair of components that generate a plane electromagnetic wave traveling in the positive z direction is (A) Ex, By.

Step-by-step solution

Analyze the given options

We need to check each of the given options to see which pair of components will create a plane electromagnetic wave traveling in the positive z direction. Let's analyze each of them: (A) Ex, By (B) Ey, Bz (C) Ex, Bz (D) Ez, Bx

Check the orthogonality

Now, let's look at each option and check if the electric and magnetic fields are orthogonal to each other: (A) Ex along the x-axis, and By along the y-axis. In this case, the electric and magnetic fields are orthogonal to each other. (B) Ey along the y-axis, and Bz along the z-axis. In this case, the electric and magnetic fields are not orthogonal to each other. (C) Ex along the x-axis, and Bz along the z-axis. In this case, the electric and magnetic fields are not orthogonal to each other. (D) Ez along the z-axis, and Bx along the x-axis. In this case, the electric and magnetic fields are orthogonal to each other.

Determine the direction of wave propagation

We will now identify the wave propagation direction for the options where the electric and magnetic fields are orthogonal to each other: (A) Ex, By: The wave propagation direction is given by the cross product of the electric field and magnetic field direction vectors: \( \vec{k} = \vec{E} \times \vec{B} \) Using the right-hand rule, we can see that the direction of wave propagation is the positive z direction. (D) Ez, Bx: The wave propagation is along the z-axis, but since the electric field is also along the z-axis, this option does not generate a plane electromagnetic wave, as the electric field should also be orthogonal to the direction of wave propagation.

Identify the correct answer

From our analysis, we can conclude that the correct answer is: (A) Ex, By: This pair of components generates a plane electromagnetic wave traveling in the positive z direction.