Electromagnetic Waves

Changing Electric Fields Produce Magnetic Fields; Maxwell’s Equations

Maxwell’s equations are the foundational laws of electromagnetism, describing how electric and magnetic fields are generated and altered by each other and by charges and currents.

• Gauss’s Law for Electricity: Relates the electric field to the electric charge enclosed by a surface.

• Gauss’s Law for Magnetism: States that there are no magnetic monopoles; the net magnetic flux through a closed surface is zero.

• Faraday’s Law of Induction: A changing magnetic field produces an electric field.

• Ampère-Maxwell Law: A magnetic field is produced by an electric current and by a changing electric field.

• Key Point: The new insight from Maxwell is that a changing electric field produces a magnetic field, even in the absence of a physical current (displacement current).

• Example: In a charging or discharging capacitor, the changing electric field between the plates creates a magnetic field, resolving inconsistencies in Ampère’s law.

Production of Electromagnetic Waves

Electromagnetic waves are self-propagating oscillations of electric and magnetic fields, generated by accelerating charges.

• Oscillating Charges: When charges accelerate (such as in an antenna), they produce changing electric and magnetic fields that propagate outward as waves.

• Wave Properties: Far from the source, electromagnetic waves can be approximated as plane waves.

• Orientation: The electric field (\(\vec{E}\)), magnetic field (\(\vec{B}\)), and direction of propagation are all mutually perpendicular.

• Speed of Propagation: Maxwell calculated the speed of electromagnetic waves in vacuum as: where \(\mu_0\) is the permeability and \(\varepsilon_0\) is the permittivity of free space.

• Result: This speed matches the measured speed of light, confirming that light is an electromagnetic wave.

Light as an Electromagnetic Wave and the Electromagnetic Spectrum

Light is a form of electromagnetic radiation, and electromagnetic waves span a broad spectrum of wavelengths and frequencies.

• Frequency-Wavelength Relationship: where \(c\) is the speed of light, \(f\) is frequency, and \(\lambda\) is wavelength.

• Electromagnetic Spectrum: Includes (in order of increasing frequency): radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.

• Example: Visible light is only a small part of the spectrum, with wavelengths from about 400 nm (violet) to 700 nm (red).

Measuring the Speed of Light

The speed of light is a fundamental constant of nature, measured with increasing precision over time.

• Historical Measurement: Early measurements (e.g., by observing Jupiter’s moons) showed that light travels at a finite speed.

• Michelson’s Experiment: Used a rotating mirror to measure the time taken for light to travel a known distance.

• Defined Value: The speed of light in vacuum is now defined as: This value is used to define the meter.

Energy in Electromagnetic Waves

Electromagnetic waves carry energy, which is stored in both their electric and magnetic fields.

• Energy Density: The total energy per unit volume is: Each field contributes half the total energy:

• Intensity: The energy transported per unit area per unit time (power per area) is called intensity: The average intensity is:

• Example: Sunlight at Earth’s surface has an average intensity of about 1000 W/m2.

Momentum Transfer and Radiation Pressure

Electromagnetic waves not only carry energy but also momentum, exerting pressure when they strike a surface.

• Radiation Pressure: The pressure exerted by electromagnetic radiation depends on whether the wave is absorbed or reflected.

  • For complete absorption:

  • For complete reflection:

• Application: Radiation pressure is significant in astrophysics (e.g., solar sails, comet tails).

Radio and Television; Wireless Communication

Radio and television signals are transmitted using electromagnetic waves, with information encoded onto a carrier wave.

• Modulation: The process of combining an audio (information) signal with a higher-frequency carrier wave.

  • Amplitude Modulation (AM): The amplitude of the carrier is varied in proportion to the signal.

  • Frequency Modulation (FM): The frequency of the carrier is varied in proportion to the signal.

• Reception: The receiving antenna picks up many frequencies; a tuner selects the desired frequency, and the signal is demodulated and amplified for output (e.g., to a loudspeaker).

• Example: FM radio provides better sound quality and noise resistance than AM radio.

Summary Table: Electromagnetic Spectrum

Key Equations

• Speed of light:

• Frequency-wavelength relation:

• Energy density:

• Intensity:

• Radiation pressure (absorbed):

• Radiation pressure (reflected):