8.5 Electromagnetic waves

Nature of electromagnetic waves

Electromagnetic waves are oscillating electric and magnetic fields that travel through space, carrying energy without needing a medium. They're responsible for everything from the light you see to the radio signals your phone picks up, and they tie together most of what you've studied in this course about electricity and magnetism.

The electric field ($\mathbf{E}$) and magnetic field ($\mathbf{B}$) in an electromagnetic wave oscillate perpendicular to each other and perpendicular to the direction the wave travels. This makes them transverse waves. Maxwell's equations provide the mathematical framework that predicts these waves and shows how changing electric fields create magnetic fields, and vice versa.

Wave-particle duality

Electromagnetic radiation behaves as both a wave and a particle, depending on the experiment. Wave behavior shows up in phenomena like interference and diffraction, where light bends around obstacles and creates patterns. Particle behavior appears in interactions with matter, such as the photoelectric effect, where light knocks electrons off a surface. Neither description alone captures the full picture.

Electromagnetic spectrum

The electromagnetic spectrum organizes all electromagnetic radiation by wavelength and frequency:

• Radio waves: longest wavelengths (used in broadcasting, communication)
• Microwaves: used in cooking, cell phone networks, radar
• Infrared: associated with heat; used in thermal imaging
• Visible light: the narrow band your eyes detect (~380–700 nm)
• Ultraviolet: causes sunburns; used in sterilization
• X-rays: penetrate soft tissue; used in medical imaging
• Gamma rays: shortest wavelengths, highest energy; emitted by radioactive nuclei

All of these travel at the same speed in vacuum but differ in wavelength and frequency.

Speed of light

The speed of light in vacuum, $c \approx 3 \times 10^8 \text{ m/s}$, is a fundamental constant. It represents the fastest speed at which energy or information can travel. Maxwell's equations predict this speed directly from two measurable constants:

$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$

where $\mu_0$ is the permeability of free space and $\epsilon_0$ is the permittivity of free space. The fact that $c$ is the same in all inertial reference frames is the starting point for Einstein's special relativity.

Maxwell's equations

Maxwell's four equations unify electricity and magnetism into a single theory. Together, they describe how charges and currents create fields, and how changing fields create each other. This self-sustaining cycle of changing $\mathbf{E}$ and $\mathbf{B}$ fields is exactly what an electromagnetic wave is.

Gauss's law

$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$

Electric field lines originate on positive charges and terminate on negative charges. The total electric flux through a closed surface is proportional to the enclosed charge. This lets you calculate electric fields for symmetric charge distributions.

Gauss's law for magnetism

$\nabla \cdot \mathbf{B} = 0$

There are no magnetic monopoles. Magnetic field lines always form closed loops; they never start or end at a point. Every magnet has both a north and south pole.

Faraday's law

$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$

A changing magnetic field induces an electric field. This is the principle behind generators and transformers. The negative sign reflects Lenz's law: the induced field opposes the change that created it.

Ampère-Maxwell law

$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

Magnetic fields are produced by electric currents ($\mathbf{J}$) and by changing electric fields. The second term, $\mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$, is Maxwell's displacement current correction. Without it, the equations wouldn't predict electromagnetic waves. This was Maxwell's key insight: a changing electric field acts like a current and generates a magnetic field.

Properties of electromagnetic waves

Wavelength and frequency

Wavelength ($\lambda$) is the distance between consecutive peaks. Frequency ($f$) is the number of cycles per second (measured in Hz). They're related by:

$c = f\lambda$

If you know one, you can find the other. Higher frequency means shorter wavelength, and it also means higher photon energy: $E = hf$, where $h$ is Planck's constant ($6.626 \times 10^{-34} \text{ J·s}$).

Amplitude and intensity

The amplitude of an electromagnetic wave is the peak value of the electric (or magnetic) field. Intensity is the power delivered per unit area, and it's proportional to the square of the amplitude:

$I \propto E_0^2$

As you move away from a point source, intensity drops according to the inverse square law: double the distance, and intensity falls to one-quarter. This is why a light bulb looks dimmer from far away.

Polarization

Polarization describes the direction in which the electric field oscillates. In linearly polarized light, $\mathbf{E}$ oscillates along a single direction. Light can also be circularly or elliptically polarized, where the direction of $\mathbf{E}$ rotates as the wave travels.

Unpolarized light (like sunlight) has electric field oscillations in random directions. A polarizing filter passes only the component along one axis, which is how polarized sunglasses reduce glare. Polarization is also used in LCD screens and 3D movie glasses.

Phase

Phase tells you where a wave is in its cycle at a given moment, measured in radians or degrees. Two waves with the same frequency can be "in phase" (peaks aligned, giving constructive interference) or "out of phase" (peaks aligned with troughs, giving destructive interference). Phase differences are central to understanding interference and diffraction patterns.

Wave propagation

Wave equation

Starting from Maxwell's equations, you can derive the electromagnetic wave equation:

$\nabla^2 \mathbf{E} = \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2}$

An identical equation holds for $\mathbf{B}$. Solutions include plane waves (uniform wavefronts extending infinitely) and spherical waves (expanding outward from a point source). The wave equation confirms that electromagnetic disturbances propagate at speed $c$.

Energy transport

Electromagnetic waves carry energy through space. The energy density stored in the fields at any point is:

$u = \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0}B^2$

For an electromagnetic wave, the electric and magnetic contributions are equal, so $u = \epsilon_0 E^2$.

Poynting vector

The Poynting vector describes the rate and direction of energy flow:

$\mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B}$

It points in the direction the wave travels, and its magnitude gives the intensity (power per unit area). The time-averaged magnitude of $\mathbf{S}$ for a sinusoidal wave is what you'd measure as the wave's intensity in practice.

Electromagnetic wave interactions

Reflection and refraction

When an electromagnetic wave hits a boundary between two media, part of it bounces back (reflection) and part enters the new medium and bends (refraction). The bending happens because the wave speed changes in different materials. Snell's law governs the angle of refraction:

$n_1 \sin\theta_1 = n_2 \sin\theta_2$

where $n$ is the refractive index of each medium. This explains how lenses focus light, why pools look shallower than they are, and how optical fibers guide light through total internal reflection.

Diffraction and interference

Diffraction is the bending of waves around obstacles or through narrow openings. It becomes noticeable when the obstacle or opening is comparable in size to the wavelength. Interference occurs when two or more waves overlap, producing regions of reinforcement (constructive) and cancellation (destructive).

The classic demonstration is the double-slit experiment: light passing through two narrow slits creates a pattern of bright and dark fringes on a screen. Diffraction gratings, which have many slits, produce sharper and more widely spaced patterns and are used in spectrometers.

Absorption and scattering

In absorption, a material takes in the wave's energy and converts it to another form (usually heat). In scattering, the wave's energy is redirected in various directions. Both depend on wavelength.

Rayleigh scattering explains why the sky is blue: shorter (blue) wavelengths scatter more than longer (red) wavelengths as sunlight passes through the atmosphere. The greenhouse effect involves absorption and re-emission of infrared radiation by atmospheric gases.

Sources of electromagnetic waves

Accelerating charges

Any accelerating electric charge emits electromagnetic radiation. A stationary charge produces only a static electric field, and a charge moving at constant velocity produces static electric and magnetic fields. But when a charge accelerates, the field "update" propagates outward as an electromagnetic wave. This is the fundamental mechanism behind all electromagnetic wave generation, from radio antennas to X-ray tubes.

Dipole radiation

An oscillating electric dipole (a charge moving back and forth) is the simplest model of an electromagnetic wave source. The radiation pattern is strongest perpendicular to the axis of oscillation and zero along the axis. Most antennas are designed around this principle, and atomic transitions that emit light can be modeled as oscillating dipoles.

Synchrotron radiation

When charged particles travel at relativistic speeds (close to $c$) along curved paths, they emit synchrotron radiation. This radiation is highly directional, very intense, and spans a broad range of the spectrum. Synchrotron facilities are used in materials science, biology, and medical imaging to produce tunable, high-brightness beams.

Applications of electromagnetic waves

Communication systems

• Radio and TV broadcasting: use radio-frequency electromagnetic waves
• Cell networks: operate at microwave frequencies (typically 700 MHz to several GHz)
• Fiber optics: transmit data as pulses of infrared light through glass fibers, enabling very high bandwidth
• Satellite communications: relay signals via microwave links for global coverage

Medical imaging

• X-ray imaging: high-energy photons pass through soft tissue but are absorbed by bone, creating contrast images
• MRI: uses radio waves and strong magnetic fields to image soft tissue without ionizing radiation
• PET scans: detect gamma rays emitted when positrons from a radioactive tracer annihilate with electrons

Note: Ultrasound uses high-frequency sound waves, not electromagnetic waves. It's often grouped with imaging techniques but operates on a completely different physical principle.

Remote sensing

• RADAR (Radio Detection and Ranging): sends microwave pulses and measures reflections to detect objects and measure distances
• LIDAR (Light Detection and Ranging): uses laser pulses for high-resolution 3D mapping
• Infrared cameras: detect thermal radiation to create heat maps, useful for night vision and building inspections
• Satellite observation: different spectral bands reveal vegetation health, ocean temperatures, atmospheric composition, and more

Electromagnetic waves in materials

Dielectrics vs conductors

In a dielectric (insulator like glass or plastic), electromagnetic waves can propagate through the material, though they slow down and may be partially absorbed. In a conductor (like a metal), free electrons respond to the wave's electric field and either reflect or absorb it. This is why metals are shiny (they reflect visible light) and why your microwave oven has a metal mesh on the door.

Permittivity and permeability

A material's permittivity ($\epsilon$) describes how it responds to electric fields, and its permeability ($\mu$) describes its response to magnetic fields. These properties determine the speed of light in the material and its refractive index:

$n = \sqrt{\epsilon_r \mu_r}$

where $\epsilon_r$ and $\mu_r$ are the relative permittivity and permeability. For most transparent materials, $\mu_r \approx 1$, so the refractive index is dominated by the electric response.

Dispersion and absorption

Dispersion means different wavelengths travel at different speeds in a material. This is why a prism splits white light into a rainbow: shorter wavelengths (violet) are slowed more than longer wavelengths (red), so they bend more.

Absorption occurs when the wave's frequency matches a natural resonance of the material. The wave's energy is transferred to the material, which is why colored glass absorbs certain wavelengths and transmits others.

Quantum aspects

Photons

At the quantum level, electromagnetic radiation comes in discrete packets called photons. Each photon carries energy:

$E = hf$

where $h = 6.626 \times 10^{-34} \text{ J·s}$ is Planck's constant and $f$ is the frequency. Higher-frequency radiation (like X-rays) consists of more energetic photons than lower-frequency radiation (like radio waves). Photons have zero rest mass but carry momentum $p = h/\lambda$.

Photoelectric effect

When light shines on a metal surface, electrons can be ejected if the photon energy exceeds the material's work function ($\phi$). The key observations are:

• Below a threshold frequency, no electrons are emitted regardless of intensity
• Above the threshold, increasing intensity increases the number of electrons but not their maximum kinetic energy
• Increasing frequency increases the maximum kinetic energy: $KE_{max} = hf - \phi$

Einstein explained this by treating light as a stream of photons, each delivering energy $hf$ to a single electron. This was strong evidence for the particle nature of light.

Compton scattering

When a high-energy photon (typically an X-ray) collides with an electron, the photon loses energy and its wavelength increases. The change in wavelength depends on the scattering angle:

$\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)$

This result can only be explained by treating the photon as a particle with definite momentum, confirming the quantum nature of light. The quantity $h/(m_e c) = 2.43 \times 10^{-12} \text{ m}$ is called the Compton wavelength of the electron.

Electromagnetic waves in modern physics

Special relativity

Einstein's special relativity is built on two postulates, one of which is that the speed of light in vacuum is the same for all inertial observers. This leads to surprising consequences like time dilation (moving clocks run slow) and length contraction (moving objects are shortened along their direction of motion). The theory also produces the mass-energy equivalence:

$E = mc^2$

Quantum electrodynamics

Quantum electrodynamics (QED) is the quantum field theory of electromagnetic interactions. It describes how photons mediate the force between charged particles and makes extraordinarily precise predictions. For example, QED predicts the anomalous magnetic moment of the electron to better than 10 significant figures, matching experiment. QED introduces concepts like virtual photons and vacuum polarization, but these go well beyond the scope of this course.