Electromagnetic waves
PHYS 301 · Maxwell's Equations
Changing electric and magnetic fields can sustain one another and propagate through space. This lesson explains electromagnetic wave structure, polarization, and energy transport.
Key equations
\vec{E}(x,t)=E_0\cos(kx-\omega t)\hat{y}\vec{B}(x,t)=B_0\cos(kx-\omega t)\hat{z}E_0=cB_0E=cB\vec{E}\times\vec{B}I=I_0\cos^2\thetac=f\lambda\nabla^2\vec{E}=\mu_0\epsilon_0\frac{\partial^2\vec{E}}{\partial t^2}\nabla^2\vec{B}=\mu_0\epsilon_0\frac{\partial^2\vec{B}}{\partial t^2}c=\frac{1}{\sqrt{\mu_0\epsilon_0}}Learning objectives
- Describe electromagnetic waves as coupled electric and magnetic fields.
- Identify the transverse structure of plane electromagnetic waves.
- Relate electric and magnetic field amplitudes in vacuum.
- Explain polarization and Malus's law.
- Recognize the electromagnetic spectrum.
Waves without a material medium
Mechanical waves need a medium, but electromagnetic waves can travel through vacuum. They consist of oscillating electric and magnetic fields that regenerate each other through Maxwell's equations.
A changing magnetic field creates a circulating electric field through Faraday's law. A changing electric field creates a circulating magnetic field through the Maxwell-Ampère law. This mutual generation allows a wave to propagate.
Plane wave structure
A simple electromagnetic plane wave traveling in the positive x-direction may have electric field in the y-direction and magnetic field in the z-direction:
ec{E}(x,t)=E_0cos(kx-omega t)hat{y}
ec{B}(x,t)=B_0cos(kx-omega t)hat{z}
The fields are perpendicular to each other and perpendicular to the direction of propagation. Electromagnetic waves are transverse waves.
Relationship between fields
In vacuum, the electric and magnetic field amplitudes are related by
More generally, at every point in a plane wave in vacuum,
The electric field, magnetic field, and propagation direction form a right-handed set. The direction of propagation is the direction of
ec{E} imes ec{B}
Polarization
Polarization describes the direction of the electric field oscillation. If the electric field stays along one fixed direction, the wave is linearly polarized. Light from many everyday sources is unpolarized, meaning it contains many polarization directions.
Polarizers transmit one component of the electric field and absorb or block the perpendicular component. If polarized light passes through an analyzer at angle , the transmitted intensity follows Malus's law:
Electromagnetic spectrum
Electromagnetic waves include radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. They differ mainly in frequency and wavelength. In vacuum,
Higher frequency means shorter wavelength and higher photon energy in quantum theory, though classical electromagnetism describes the wave fields themselves.
Production of electromagnetic waves
Accelerating charges produce electromagnetic radiation. In an antenna, charges oscillate back and forth, creating changing electric and magnetic fields that detach and propagate outward. Thermal radiation is produced by accelerating charges in matter due to microscopic motion.
Atoms and nuclei can also emit electromagnetic radiation through quantum transitions.
Wave equation
In empty space, Maxwell's equations imply that electric and magnetic fields satisfy wave equations:
abla^2 ec{E}=mu_0epsilon_0 rac{partial^2 ec{E}}{partial t^2}$$abla^2 ec{B}=mu_0epsilon_0 rac{partial^2 ec{B}}{partial t^2}$$
These equations show that electromagnetic disturbances propagate at speed
c= rac{1}{sqrt{mu_0epsilon_0}}
The big idea
Electromagnetic waves are self-propagating oscillations of electric and magnetic fields. They are transverse, can travel through vacuum, carry energy and momentum, and span a vast spectrum from radio waves to gamma rays. Maxwell's equations show that light itself is an electromagnetic wave.
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