Premed · Premed · Physics 2
Lecture 14: Electromagnetic Waves
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- State Maxwell's equations qualitatively and explain how they predict electromagnetic waves
- Describe the properties of electromagnetic waves (transverse, self-propagating, speed of light)
- Identify the regions of the electromagnetic spectrum and their applications
- Calculate the energy, intensity, and radiation pressure of electromagnetic waves
- Explain the Poynting vector and its physical significance
Lecture Content
I. Maxwell's Equations and the Displacement Current
Maxwell's four equations unify electricity and magnetism into a single, coherent framework. The first equation, Gauss's law for E, states that the flux of the electric field through a closed surface equals the enclosed charge divided by epsilon_0. The second, Gauss's law for B, states that the magnetic flux through any closed surface is zero, reflecting the absence of magnetic monopoles. The third, Faraday's law, states that a changing magnetic flux induces an electric field around a closed loop. The fourth, the Ampere-Maxwell law, states that a magnetic field can be produced both by a current and by a changing electric flux.
Maxwell's key contribution was the displacement current term mu_0 epsilon_0 d(Phi_E)/dt in the fourth equation. This term means that a changing electric field produces a magnetic field even in the absence of a physical current. It completes a beautiful symmetry: just as a changing magnetic field produces an electric field (Faraday's law), a changing electric field produces a magnetic field. This reciprocal relationship allows electromagnetic disturbances to sustain themselves and propagate through space without needing any medium. A changing E creates B, which itself changes and creates E, and so on, producing a self-propagating electromagnetic wave.
II. Properties of Electromagnetic Waves
Electromagnetic waves are transverse waves in which E and B oscillate perpendicular to the direction of propagation and perpendicular to each other. The direction of propagation is given by E x B. In vacuum, the speed of electromagnetic waves is c = 1/sqrt(mu_0 epsilon_0) = 2.998 x 10^8 m/s. Maxwell's calculation of this speed, and his recognition that it equaled the measured speed of light, led to his prediction that light itself is an electromagnetic wave.
The magnitudes of E and B are related by E = cB at all points and times. Electromagnetic waves satisfy the standard wave equation c = f lambda, where f is the frequency and lambda is the wavelength. All electromagnetic waves travel at c in vacuum, regardless of their frequency. In a medium with permittivity epsilon and permeability mu, the wave speed decreases to v = 1/sqrt(mu epsilon) < c, and the index of refraction is n = c/v.
<image>A three-dimensional diagram of an electromagnetic wave propagating in the x-direction. The electric field E oscillates sinusoidally in the y-direction (vertical plane), and the magnetic field B oscillates sinusoidally in the z-direction (horizontal plane). E and B are in phase (peaks and troughs aligned), perpendicular to each other, and both perpendicular to the propagation direction. The wavelength lambda is labeled as the distance between successive peaks. Arrows indicate E x B points in the direction of propagation (+x). The relationship E_max = cB_max is noted.</image>
III. The Electromagnetic Spectrum
Electromagnetic waves span an enormous range of frequencies and wavelengths, yet they are all fundamentally the same phenomenon, differing only in frequency. Listed in order of increasing frequency and decreasing wavelength, the regions of the spectrum are as follows.
Radio waves (lambda > 1 mm, f < 300 GHz) are used for AM/FM radio, television, cell phones, and Wi-Fi. Microwaves (lambda approximately 1 mm to 30 cm, f approximately 1 GHz to 300 GHz) are used in microwave ovens, radar, and satellite communications. Infrared radiation (lambda approximately 700 nm to 1 mm) is associated with thermal radiation, remote controls, IR spectroscopy, and thermal imaging. Visible light (lambda approximately 400-700 nm) spans from violet at 400 nm to red at 700 nm and is the only portion of the spectrum detected by the human eye. Ultraviolet radiation (lambda approximately 10-400 nm) causes sunburn, promotes vitamin D synthesis, and is used for sterilization and fluorescence. X-rays (lambda approximately 0.01-10 nm) are used in medical imaging, crystallography, and CT scans. Gamma rays (lambda < 0.01 nm) are produced by nuclear reactions and are used in cancer treatment, PET scans, and sterilization.
<image>The electromagnetic spectrum displayed as a horizontal band, with wavelength increasing to the left and frequency increasing to the right. Each region is labeled and color-coded: radio (red), microwave (orange), infrared (dark red), visible light (expanded into the rainbow: red, orange, yellow, green, blue, violet), ultraviolet (purple), X-rays (blue-gray), gamma rays (dark blue). Below each region, approximate wavelength ranges and example applications are listed. A logarithmic scale for wavelength (from 10^3 m to 10^-14 m) and frequency (from 10^5 Hz to 10^22 Hz) runs along the bottom.</image>
IV. Energy and Intensity of EM Waves
Electromagnetic waves carry energy, with contributions from both the electric and magnetic fields. The electric energy density is u_E = (1/2) epsilon_0 E^2, and the magnetic energy density is u_B = B^2 / (2 mu_0). In an electromagnetic wave, these two contributions are always equal, so the total energy density is u = epsilon_0 E^2 = B^2/mu_0.
The Poynting vector S = (1/mu_0) E x B describes the rate of energy flow per unit area. It points in the direction of wave propagation, and its magnitude is S = EB/mu_0 = E^2/(mu_0 c) = cB^2/mu_0, measured in W/m^2.
The intensity I is the time-averaged power per unit area: I = S_avg = E_max^2 / (2 mu_0 c) = c epsilon_0 E_max^2 / 2. For a point source emitting total power P, the intensity at distance r follows the inverse square law: I = P / (4 pi r^2).
V. Radiation Pressure and Momentum
Electromagnetic waves carry momentum as well as energy. The momentum per unit volume is u/c = S/c^2. The radiation pressure on a surface that perfectly absorbs the radiation is P_rad = I/c, and on a surface that perfectly reflects it, the pressure doubles to P_rad = 2I/c.
Although radiation pressure is small for ordinary light, it produces measurable effects on astronomical scales and in precision experiments. It is responsible for comet tails being pushed away from the sun, it enables solar sails for spacecraft propulsion, it allows optical tweezers to trap and manipulate microscopic objects with focused laser beams, and inside stars, radiation pressure balances gravitational collapse to maintain stellar equilibrium.
VI. Production and Reception of EM Waves
Electromagnetic waves are produced by accelerating charges. An oscillating charge creates oscillating E and B fields that propagate outward as a wave. The radiated power is proportional to the square of the acceleration (the Larmor formula).
In practical antenna radiation, an oscillating current in an antenna such as a half-wave dipole produces electromagnetic waves. The radiation pattern depends on the antenna geometry, with maximum radiation emitted perpendicular to the antenna axis. Reception works in reverse: the oscillating electric field of an incoming wave drives currents in a receiving antenna, and resonant RLC circuits select a specific frequency.
Medical applications of electromagnetic radiation are extensive. X-rays are used for diagnostic imaging, gamma rays for radiation therapy and PET scans, microwaves for diathermy (deep tissue heating), infrared for thermal therapy, and ultraviolet for phototherapy of skin conditions.
<image>A diagram showing EM wave production by an oscillating electric dipole antenna. The antenna (vertical rod) has charges oscillating up and down at frequency f. Concentric wavefronts expand outward from the antenna. The radiation pattern is shown as a donut-shaped (toroidal) surface around the antenna, with maximum intensity perpendicular to the antenna axis and zero intensity along the axis. At a distant point, the E field oscillates vertically and the B field oscillates horizontally, both perpendicular to the radial direction of propagation.</image>
