Problem 2197 - JEE Physics

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) \(E z, B x\)

Step-by-Step Solution

Answer

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

Step 1: 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

Step 2: 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.

Step 3: 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.

Step 4: 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.

Key Concepts

Electric Field

The electric field is a crucial part of understanding electromagnetic waves. It represents the region around a charged particle where a force would be exerted on other charged particles. In the context of electromagnetic waves, such as the one described in the textbook exercise, the electric field forms part of the wave's oscillating components.

The electric field vector in a wave travels perpendicular to both the magnetic field component and the direction of wave propagation. This orthogonality is essential because it dictates the formation and propagation of electromagnetic waves. In the given problem, components such as \( E_x \) and \( E_y \) refer to the electric field's alignment along specific Cartesian coordinates.

• The electric field in a wave is vectorial, meaning it has both magnitude and direction.
• For a wave traveling in the z-direction, the electric field must be perpendicular to \( z \) for effective wave propagation.
• The direction of the electric field defines the polarization of the wave.

Understanding how these vectors interact helps to determine the wave's characteristics, such as propagation direction.

Magnetic Field

The magnetic field is another essential part of electromagnetic waves, interacting closely with the electric field. In an electromagnetic wave, the magnetic field is perpendicular to both the electric field and the direction of wave propagation. This arrangement ensures the wave can efficiently travel through space.

The magnetic component (\( \vec{B} \)) discussed in the problem, like \( B_x \) and \( B_y \), denotes its orientation in a coordinate system.

• Magnetic fields are generated by electric currents or changing electric fields, which is why they are so integral to electromagnetic waves.
• The strength and direction of these fields can define how and where the wave travels.
• Just like the electric field, the magnetic field's direction is crucial for determining wave properties.

In the exercise, recognizing the orthogonality between the electric and magnetic fields helps identify potential wave propagation paths. In option (A), the fields \( E_x \) and \( B_y \) being perpendicular allows for propagation in the desired direction.

Wave Propagation

Wave propagation refers to the movement of waves through a medium or space. For electromagnetic waves, this involves the interaction and continuous creation of electric and magnetic fields at right angles.

In the exercise, determining which components allow the wave to move in the positive z-direction requires understanding this concept. The right-hand rule is often utilized:

• Place fingers of the right hand along the direction of the electric field.
• Rotate your palm towards the magnetic field direction.
• Your thumb then points in the direction of wave propagation.

For a wave to propagate efficiently, the electric and magnetic fields need to be perpendicular not only to each other but also to the direction the wave is traveling. This orthogonality ensures the wave's energy is transferred smoothly across space. Criteria such as these allow us to determine that option (A), \( E_x \) and \( B_y \), accurately captures the idea of wave propagation in the z-direction as posed by the problem.