← Physics II: E&M and Waves

Unit 12: Maxwell's Equations and Electromagnetic Waves

Four equations that finally say everything electricity and magnetism can do, and the wave they predict: light, carrying energy and momentum through empty space.

8 topics · 31 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Assembling Maxwell's Equations

  1. How it worksHow do Gauss's law for electric fields and Gauss's law for magnetic fields each describe a different kind of source?
  2. How it worksWalk me through how Faraday's law and the Ampere-Maxwell law each describe one field curling because the other changes.
  3. Ties togetherHow do Gauss's law, Faraday's law and Ampere's law assemble into Maxwell's four equations?
  4. Going furtherWhat happens to the symmetry between the electric and magnetic equations given that magnetic monopoles have never been found?
Your turn1 of 2
Coulomb's law alone can describe the field of any static charge distribution. Why wasn't a single law like that enough to describe every electromagnetic phenomenon?

The Displacement Current

  1. How it worksHow does Maxwell's displacement current fix Ampere's law so it still works even where a circuit has a gap?
  2. Worked exampleFind the displacement current between two 0.01 square meter capacitor plates whose field changes at 5 times 10 to the 11 volts per meter per second.
  3. Common mistakeWhy isn't the displacement current an actual flow of charge the way an ordinary conduction current is?
  4. Going furtherWhat happens to the magnetic field pattern around a capacitor's gap once the displacement current is included?
Your turn1 of 2
Find the displacement current between two 35 square centimeter capacitor plates whose field changes at 6 times 1011 volts per meter per second. Use epsilon0 = 8.85x10−12 C2/(N m2).

Plane Electromagnetic Waves

  1. How it worksHow do Faraday's law and the Ampere-Maxwell law let a changing E field regenerate a changing B field, endlessly?
  2. Worked exampleFind the wavelength of an electromagnetic wave with a frequency of 100 megahertz.
  3. How it worksWalk me through why the E and B fields in a plane wave stay perpendicular to each other and to the direction of travel, oscillating in phase.
  4. Ties togetherHow does this self-sustaining wave connect back to the induced fields you met in Faraday's and Ampere's laws?
Your turn1 of 2
Find the wavelength of an electromagnetic wave with a frequency of 699 megahertz. Use c = 3x108 m/s.

The Speed of Light from Maxwell's Equations

  1. How it worksHow do the electric and magnetic constants of empty space combine to set the speed of an electromagnetic wave?
  2. Worked exampleFind the speed of an electromagnetic wave in vacuum given permittivity 8.85 times 10 to the -12 and permeability 4 π times 10 to the -7 in SI units.
  3. Going furtherWhat happens to an electromagnetic wave's speed the moment it travels through a medium instead of a vacuum?
Your turn1 of 2
Find the speed of an electromagnetic wave in a medium with relative permittivity 7 and relative permeability 4. Use c = 3x108 m/s.

Energy and the Poynting Vector

  1. How it worksHow does the Poynting vector combine a wave's electric and magnetic fields into its energy flow direction and rate?
  2. Worked exampleFind the intensity of an electromagnetic wave with an electric field amplitude of 200 volts per meter.
  3. Common mistakeWhy doesn't an electromagnetic wave's energy live only in its electric field, with the magnetic field just along for the ride?
  4. Going furtherWhat happens to a wave's intensity as its electric field amplitude is doubled?
  5. How it worksHow does the Poynting vector's direction always point along the direction the electromagnetic wave itself travels?
Your turn1 of 2
Find the intensity of an electromagnetic wave with an electric field amplitude of 699 volts per meter. Use epsilon0 = 8.85x10−12 C2/(N m2) and c = 3x108 m/s.

Radiation Pressure

  1. How it worksHow does an electromagnetic wave carrying momentum let it exert pressure on whatever absorbs or reflects it?
  2. Worked exampleFind the radiation pressure exerted by a fully absorbed wave with intensity 1000 watts per square meter.
  3. Going furtherWhat happens to the radiation pressure on a surface if it reflects incoming light perfectly instead of absorbing it?
Your turn1 of 2
Write the radiation pressure of a fully absorbed electromagnetic wave as a function of I, its intensity in watts per square meter. Use c = 3x108 m/s.

Polarization of Light

  1. How it worksHow does a polarizing filter only let through the component of a light wave's electric field aligned with its own axis?
  2. Worked exampleFind the transmitted intensity when 80 watt per square meter unpolarized light passes through one polarizer, then a second one set at 30 degrees.
  3. Common mistakeWhy doesn't ordinary sunlight already arrive with its electric field pointing in just one fixed direction?
  4. Going furtherWhat happens to the transmitted intensity as the angle between two polarizers grows from 0 to 90 degrees?
  5. Worked exampleFind the angle between two polarizers that cuts an initially polarized beam's intensity exactly in half.
Your turn1 of 2
Find the transmitted intensity when 142 watt per square meter unpolarized light passes through one polarizer, then a second one set at 55 degrees to the first.

The Electromagnetic Spectrum

  1. How it worksHow does the electromagnetic spectrum organize every kind of light purely by its wavelength and frequency?
  2. Common mistakeWhy doesn't a gamma ray travel through vacuum any faster than an ordinary radio wave does?
  3. Ties togetherHow does the electromagnetic spectrum connect the speed-of-light relation to every named band of light there is?
Your turn1 of 2
A gamma ray and a radio wave are both traveling through vacuum. How do their speeds compare?