Suggested Homework Problem For Study

Suggested Problem For Homework 6in This Problem We Study The Formation

Analyze the formation of standing electromagnetic waves between two perfectly reflecting conducting infinite planes separated by distance L. Assume vacuum between the planes, with two sinusoidal plane waves traveling in opposite directions, described by specific equations for electric and magnetic fields. Address questions related to the resultant fields, nodal and anti-nodal planes, phase relationships, and the motion of a small charge placed within the standing wave pattern. Express all answers in terms of the given quantities and fundamental constants.

Paper For Above instruction

In this analysis, we explore the nature of standing electromagnetic waves formed between two perfect reflecting conducting planes. These planes create a confined space where electromagnetic waves reflect back and forth, resulting in a stationary wave pattern characterized by nodes and anti-nodes. We begin by considering two sinusoidal plane waves traveling in opposite directions along the x-axis, each described by their respective electric and magnetic field equations. The wave traveling in the positive x-direction is given by equations involving the sinusoidal functions for electric field \(\mathbf{E}\) and magnetic field \(\mathbf{B}\), with the wave vector \(k=\frac{\pi}{\lambda}\) and angular frequency \(\omega\). The wave moving in the negative x-direction has corresponding expressions with appropriate phase differences.

(A) Total Electric and Magnetic Fields:

The superposition principle allows us to find the total fields within the cavity by summing the individual waves. The electric field \(\mathbf{E}_\text{total}(x,t)\) is obtained by adding the fields of both waves, resulting in a cosine modulation that depends on both position and time. Similarly, the total magnetic field \(\mathbf{B}_\text{total}(x,t)\) results from the sum of individual magnetic components, involving sinusoidal functions. The combined expressions reveal standing wave patterns with specific spatial and temporal variations, with electric and magnetic fields oscillating in phase or out of phase depending on their positions.

(B) Nodal and Anti-Nodal Planes of Electric Field:

Nodal planes for the electric field are the positions where the total electric field remains zero at all times, corresponding to destructive interference points. Anti-nodal planes are where the electric field reaches maximum amplitude, corresponding to constructive interference. The positions of these planes are determined by the boundary conditions imposed by the reflecting planes and the superposition of waves, which depend on the wavelength \(\lambda\), the phase differences, and the standing wave formation conditions. The total number of nodal or anti-nodal planes is influenced by the cavity length \(L\), the wavelength \(\lambda\), and the modes supported by the boundary conditions.

(C) Nodal and Anti-Nodal Planes of Magnetic Field and Their Correspondence:

Similarly, the magnetic field exhibits its own set of nodal and anti-nodal planes, which often coincide with or are offset relative to those of the electric field. In ideal electromagnetic standing waves, the electric and magnetic fields are in phase, leading to corresponding nodes and anti-nodes at specific locations along the x-axis. The correspondence arises because the electric and magnetic fields are coupled in Maxwell's equations, sustaining a phase relationship that ensures the standing wave pattern maintains consistency between both fields.

(D) Phase Relationship and Zero Electric Field Conditions:

At times when the total electric field component is zero everywhere, the magnetic field component reaches its maximum or minimum, indicating that the magnetic field is in phase opposition or in a specific phase relation with the electric field. This phase relationship is characteristic of standing electromagnetic waves, where electric and magnetic fields are often out of phase in time but coupled spatially. In contrast, traveling electromagnetic waves exhibit a \(\frac{\pi}{2}\) phase difference between electric and magnetic fields, ensuring they are in phase spatially but out of phase temporally.

(E) Motion of a Small Charge in the Standing Wave:

Placing a small positive charge \(q\) at specified points within the standing wave pattern results in forces due to the electric field. If placed at a node of the electric field, the charge experiences no net electrostatic force and may remain stationary or oscillate based on other factors. At an anti-node, the charge experiences maximum oscillatory force in the direction of the electric field, leading to driven oscillations that depend on the position relative to the standing wave pattern. The dynamics differ at distances \(d_L = 0.25\lambda\) and \(d_L=0.75\lambda\), with forces and subsequent motion being explained based on the amplitude variation of the electric field at those points.

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