Question

A left-circularly polarized plane wave is normally incident onto the surface of a perfect conductor. (a) Construct the superposition of the incident and reflected waves in phasor form. (b) Determine the real instantaneous form of the result of part ( \(a\) ). \((c)\) Describe the wave that is formed.

Short answer

Question: Describe the wave formed by the superposition of an incident and reflected left-circularly polarized plane wave on a perfect conductor surface. Answer: The resulting wave is a left-circularly polarized standing wave pattern along the z-axis, which forms due to the interference between the incident and reflected waves. Its intensity will depend on the distance from the conductor's surface. The electric field is zero at the conductor's surface, as required by the perfect conductor boundary condition.

Step-by-step solution

1. Incident wave in phasor form

Since we are working with a left-circularly polarized plane wave, we can represent the incident wave as a superposition of two linearly polarized waves with a phase difference of \(-\pi/2\). The phasor form for the incident wave is given by: $$\vec{E_i} = E_0 (\hat{x} + j\hat{y}) e^{-j k z}$$ where \(E_0\) is the amplitude of the wave, \(j\) is the imaginary unit, \(k\) is the wave number, and \(z\) is the coordinate along the direction of wave propagation.

2. Reflecting the incident wave

When the wave encounters a perfect conductor, the electric field at the conductor's surface is forced to be zero. This means that the reflected wave will have the same amplitude but opposite phase to make the total electric field zero at the surface: $$\vec{E_r} = E_0 (\hat{x} - j\hat{y}) e^{-j k (2d - z)}$$ where \(d\) is the distance from the conductor's surface.

3. Adding the reflected wave to the incident wave

Now, let's add the incident and reflected waves to find the total electric field in phasor form: $$\vec{E_T} = \vec{E_i} + \vec{E_r} = E_0 (\hat{x} + j\hat{y}) e^{-j k z} + E_0 (\hat{x} - j\hat{y}) e^{-j k (2d - z)}$$

4. Convert the phasor form to real instantaneous form

To obtain the real instantaneous form, we take the real part of the phasor form: $$\vec{E}(z, t) = Re[\vec{E_T} e^{j \omega t}] = E_0 (\hat{x} \cos(kz - \omega t) + \hat{y} \sin(kz - \omega t)) + E_0 (\hat{x} \cos(k(2d - z) - \omega t) - \hat{y} \sin(k(2d - z) - \omega t)) $$

5. Describe the resulting wave

The resulting wave is the superposition of the incident and reflected left-circularly polarized plane waves. The waves interfere with each other to create a standing wave pattern along the \(z\)-axis. The wave is still left-circularly polarized, but its intensity will depend on the distance from the conductor's surface. At the conductor's surface, the electric field is zero as required by the perfect conductor boundary condition.

Key concepts

Perfect Conductor

When we talk about a perfect conductor in the context of electromagnetic waves, we're describing an idealized material that has an infinite electrical conductivity. In such materials, the electric field inside the conductor is zero, because charges redistribute themselves instantly in response to an external electric field to negate its effect.

In the case of an electromagnetic wave approaching a perfect conductor, as seen in our exercise, the wave cannot penetrate the conductor. Instead, it gets reflected, and the perfect conductor acts as a perfect mirror for the wave. This reflection creates interesting patterns, such as standing waves, which are a result of the superposition of the incident and reflected waves. The boundary conditions at the surface of the conductor dictate that the tangential component of the electric field must be zero, which leads to the reflected wave having an equal amplitude but opposite phase to the incident wave.

Phasor Form

The phasor form of a wave is a complex representation that makes it easier to deal with sinusoidal functions, especially when solving problems involving interference, superposition, and reflections. In phasor notation, a wave is represented as a complex exponential function, where the magnitude gives the amplitude and the angle represents the phase angle.

This method turns differential equations into algebraic ones, greatly simplifying the analysis of the wave, particularly when you're combining waves to see their collective effect on a point in space. In the exercise, the incident wave's electric field is given in phasor form to easily combine it with the reflected wave and predict the outcome of their interaction when normally incident on a perfect conductor.

Standing Wave Pattern

A standing wave pattern is a stationary pattern of oscillations that results when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. Unlike traveling waves, which carry energy away from their source, standing waves have nodes, where the medium doesn't move, and antinodes, where the medium moves with maximum amplitude.

In the context of the exercise, the reflective properties of a perfect conductor cause the incident electromagnetic wave to superpose with its reflected counterpart, leading to the formation of a standing wave pattern. This pattern's characteristics are determined by the interference of the incident and reflected waves, creating regions where the electric field intensity varies along the space in front of the conductor.

Electromagnetic Wave Reflection

Electromagnetic wave reflection occurs when a wave encounters a boundary or change in medium, causing the wave to bounce back into the original medium. Reflection is governed by the electromagnetic properties of the media involved. In the case of a perfect conductor, all the incident energy is reflected, resulting in no transmission into the conductor.

The exercise demonstrates the reflection of a left-circularly polarized plane wave upon a perfect conductor. Here, reflection results in a phase change, which is incorporated into the reflected wave as a negative of the incident wave's complex representation in phasor form. This is a critical aspect that helps establish a standing wave due to the constructive and destructive interference.

Boundary Conditions

Boundary conditions in electromagnetism are rules that the electric and magnetic fields must obey at the interfaces between different materials. They're derived from Maxwell's equations and are fundamental to determining the behavior of electromagnetic fields at boundaries. Key boundary conditions state that the tangential component of the electric field and the normal component of the magnetic flux density must be continuous across a boundary.

In the case of a perfect conductor, these conditions lead to specific outcomes like the requirement that the tangential electric field must be zero at the surface. Conforming to these conditions, as shown in the given exercise solution, affects the form of the reflected wave and ensures that the resulting electric field satisfies the physical constraints imposed by the boundary.