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Chapter 24: Problem 18

Given the plane wave \(\mathbf{E}=\gamma[(3+2 i) \hat{\mathbf{x}}+\) \((3+4 i) \hat{\mathbf{y}}] e^{i(k z-\omega i)}\) where \(\gamma\) is a positive constant and \(k\) is positive and real. Find: \(E_{1}, E_{2}, \vartheta_{1}, \vartheta_{2}, \Delta\), and the angle \(\varphi\) between the major axis of the ellipse and the \(x\) axis. Does the wave have positive or negative helicity? Is its sense of polarization right-handed or left-handed?

Short Answer

Expert verified
The magnitudes are \(E_1=|\gamma (3 + 2i)|\) and \(E_2 = |\gamma (3 + 4i)|\), the phase angles are \(\vartheta_1 = \arg(\gamma (3 + 2i))\) and \(\vartheta_2 = \arg(\gamma (3 + 4i))\), phase difference is \(\Delta = \vartheta_1 - \vartheta_2\), the angle \(\varphi = 1/2 \atan(2 \Delta / (E_1^2 - E_2^2))\). The polarization helicity is positive and the sense of polarization is left-handed.

Step by step solution

01

Calculating Magnitudes

The magnitudes of electric field components are \(E_1=|\gamma (3 + 2i)|\) and \(E_2 = |\gamma (3 + 4i)|\).
02

Computing Phases

The phases are given by the arguments of the complex numbers: \(\vartheta_1 = \arg(\gamma (3 + 2i))\) and \(\vartheta_2 = \arg(\gamma (3 + 4i))\).
03

Computing Phase Difference

The phase difference is obtained by \(\Delta = \vartheta_1 - \vartheta_2\).
04

Computing Angle Between Major Axis and x-axis

The angle between the major axis of the ellipse and the x-axis, \(\varphi\), is given by \(\varphi = 1/2 \atan(2 \Delta / (E_1^2 - E_2^2))\).
05

Polarization Helicity and Sense

To find out the rotation sense of the polarization and the helicity of the wave, we look at the exponent in the wave function. As the phase decreases with increasing z (i.e., waves are moving in the +z direction), the wave has positive helicity. Because the y component leads the x component by \(\pi/2\), the sense of the polarization is left-handed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polarization
Polarization describes the orientation of oscillations in a wave, such as an electromagnetic wave. In the context of our plane wave example, polarization indicates how the electric field components are oriented as the wave moves through space.

For elliptically polarized waves, the electric field traces out an ellipse in the plane perpendicular to the direction of propagation.
  • In our wave, the complex coefficients \(3+2i\) and \(3+4i\) determine the shape and orientation of this ellipse.
  • The angle \(\varphi\) between the major axis of the ellipse and the x-axis can be calculated to describe the wave's polarization state.
A right-hand sense of polarization means that as you look along the direction of propagation, the electric field vector rotates in a counterclockwise direction. Conversely, a left-hand polarization denotes a clockwise rotation.

In this example, due to the phase relationships, the polarization sense is left-handed.
Complex Numbers
Complex numbers play a crucial role in understanding electromagnetic waves. They provide a convenient way to represent oscillations and phases.

In our example, the wave has components \(3+2i\) and \(3+4i\), where:
  • The real part corresponds to the amplitude.
  • The imaginary part defines the phase shift.
The magnitude of a complex number \(a + bi\) is given by \(|a+bi| = \sqrt{a^2 + b^2}\). It tells us the maximum possible value of the wave's electric field.

The phase or argument of a complex number is derived using \(\arg(a+bi) = \tan^{-1}(b/a)\). This determines the position of the oscillation in its cycle at a given point.

In electromagnetic waves, these properties of complex numbers help in calculating magnitudes and phase differences, essential for understanding wave behavior.
Wave Propagation
Wave propagation refers to the movement of waves through space and can be affected by the wave's frequency, medium, and direction.

Our wave propagates in the z-direction, with the exponential term \(e^{i(kz-\omega t)}\). Here:
  • \(k\) 鈥� the wave number, indicating how many oscillations occur per unit distance.
  • \(\omega\) 鈥� the angular frequency, representing how fast the wave oscillates over time.
As the wave moves in the +z direction, these parameters govern how quickly the wave's peaks and troughs pass through a given point.

In practical applications, understanding wave propagation helps in designing communication systems, antennas, and other devices relying on wave transmission.
Helicity
Helicity is a property related to the spin and propagation direction of a wave. It defines whether the wave's rotational sense aligns with its direction of propagation.

For helicity:
  • A wave with positive helicity has its rotational sense in the same direction as its motion.
  • A negative helicity means the rotation is opposite to the wave's movement.
In our example, as the phase decreases with increasing z, the wave exhibits positive helicity. This means that its rotational (angular) momentum direction aligns with the wave's propagation.

Helicity is an important concept in fields like quantum mechanics and in analyzing circularly polarized waves, influencing their interactions and effects in various media.

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Most popular questions from this chapter

Consider a plane wave that is a superposition of two independent plane waves with parallel electric fields and has the form \(\mathbf{E}=\) \(\hat{\mathbf{x}} E_{\alpha} e^{i\left(k z-\omega t+\vartheta_{\alpha}\right)}+\hat{\mathbf{x}} E_{\beta} e^{i\left(k z-\omega t+\vartheta_{\beta}\right)}\) where \(k\) is real. Find \(\langle\mathbf{S}\rangle\) and show that it is not equal to the sum of the average Poynting vectors for each component. (This is a result of " interference.")

Find the ratio \(\left\langle u_{m}\right\rangle /\left\langle u_{e}\right\rangle\) for a plane wave in a conducting medium. Then find the approximate expressions for this ratio for the limiting cases of an insulator and a good conductor.

As was noted after \((24-53)\), a superposition of plane waves traveling in a dispersive medium generally changes its form as it progresses. As an extreme example, consider two waves of equal real amplitudes and with almost the same propagation constants and frequencies that are traveling in the positive \(z\) direction, so that their sum has the form $$ \begin{aligned} \psi=& \psi_{0} e^{i[(k+d k) z-(\omega+d \omega) t]} \\ &+\psi_{0} e^{i[(k-d k) z-(\omega-d \omega) t]} \end{aligned} $$ Show that $$ \begin{aligned} \operatorname{Re} \psi &=2 \psi_{0} \cos [(d k) z-(d \omega) t] \cos (k z-\omega t) \\ &=\psi_{m} \cos (k z-\omega t) \end{aligned} $$ where \(\psi_{m}\) is called the " modulation." Thus (24-143) has the form of a wave with average (24-143) has the form of a wave with average propagation constant \(k\) and average frequency \(\omega\). Its amplitude is not constant, but itself is a wave and travels with the group velocity \(v_{G}\) $$ v_{G}=\frac{d \omega}{d k} $$ What is the spatial period (wavelength) \(\lambda\) of the main wave? What is the wavelength \(\lambda_{m}\) of \(\psi_{m}\) ? Find the ratio \(\lambda_{m} / \lambda\). Sketch (24-143) at \(t=0\), and at a slightly later time, and thus verify that the form of \(\operatorname{Re} \psi\) has changed. Identify which physical feature of your sketch travels with the phase velocity \(v=\omega / k\) and which with the group velocity \(v_{G}\). Why do you think the name "group" velocity was given to (24-144)? (This specific result is an example of the more general phenomenon known as beats.)

A particle of charge \(q\) and mass \(m\) is traveling with velocity \(\mathbf{u}\) in the field of a plane electromagnetic wave in free space that itself is traveling in the \(z\) direction. Find the force on the particle. What does this become for the special case in which the particle is traveling in the same direction as the wave? What is the direction of the force? Under what conditions (if any) will the force vanish in this case?

Show that the electric field of a righthanded circularly polarized wave can be written in the form \(\mathbf{E}_{+}=E_{0}(\hat{\mathbf{x}}-i \hat{\mathbf{y}}) e^{i(k z-\omega t+\vartheta)}\). What is the corresponding expression if it is left-handed?
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