Electromagnetic Waves: Energy, Momentum, and Production

Learning Objectives

• Understand the energy and momentum carried by electromagnetic (EM) waves.

• Define and calculate radiation intensity and radiation pressure.

• Describe how EM waves are produced, including the role of antennas and oscillating dipoles.

EM Waves: Basic Properties

Characteristics of Electromagnetic Waves

• Electromagnetic waves have no mass.

• They travel at the speed of light, c, in a vacuum ( m/s).

• EM waves carry both energy and momentum.

Energy Transfer in EM Waves

Key Quantities and Relations

The following table summarizes important quantities related to energy transfer in EM waves:

Poynting Vector

Definition and Physical Meaning

• The Poynting vector () represents the rate of energy transfer per unit area by an EM wave.

• It is defined as:

(W/m2)

• points in the direction of wave propagation.

• The energy carried by EM waves is stored in both the electric () and magnetic () fields.

Energy Density in EM Waves

• The total energy density is the sum of the energy densities of the electric and magnetic fields:

• For a plane wave, .

Key Equations for the Poynting Vector

• Magnitude of the Poynting vector:

• Relationship between and in a plane wave:

EM Wave Intensity

Definition and Calculation

• Intensity () is the time-averaged value of the Poynting vector:

(W/m2)

• For a sinusoidal wave:

• For an isotropic source, intensity decreases with the square of the distance from the source:

• For a plane wave, intensity is constant.

Example: Light Bulb Problem

• A 100 W light bulb radiates EM energy isotropically. If 10% of the power is in the visible spectrum, what is the intensity 2 m from the bulb?

• Solution steps:

  1. Visible power:

  2. Area at 2 m:

  3. Intensity:

Radiation Pressure

Definition and Equations

• EM waves exert radiation pressure when they strike a surface due to their momentum.

• For complete absorption:

• For complete reflection:

• Where is the intensity and is the speed of light.

Example: NASA Solar Sail

• Given: Area = 80 m2, mass = 30 kg, solar constant = 1360 W/m2.

• Find: Pressure on the sail, acceleration, and distance traveled in a year.

• Pressure: (for absorption), (for reflection).

• Acceleration:

• Distance in time : (if starting from rest).

Production of Electromagnetic Waves

Accelerating Charges

• EM waves are produced by accelerating electric charges.

• A stationary charge produces a constant electric field.

• A charge moving at constant velocity (current) produces a stationary magnetic field.

• Only accelerating charges produce propagating EM waves.

Additional info: The detailed calculation of the fields from accelerating charges involves advanced concepts such as Green's functions and retarded potentials.

Oscillating Electric Dipole

• An oscillating electric dipole is a classic source of EM waves.

• As the dipole oscillates, it creates time-varying electric and magnetic fields that propagate outward as EM waves.

Antennas

Dipole and Linear Antennas

• An antenna is a device designed to efficiently produce or receive EM waves.

• The simplest antenna is the half-wave (Hertz) dipole, which operates efficiently from 3 kHz to 3 GHz.

• The E and B fields from a dipole antenna are those of an oscillating dipole.

Other Types of Antennas

• There are many types of antennas for different applications, including parabolic, loop, and monopole antennas.

• All antennas function by creating or detecting oscillating electric and magnetic fields.

Electromagnetic Wave Spectrum

Overview

• All EM waves, from radio waves to gamma rays, are described by Maxwell's equations.

• The EM spectrum includes (in order of increasing frequency): radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.

Additional info: The visible spectrum is only a small part of the entire EM spectrum, which spans many orders of magnitude in wavelength and frequency.