Maxwell's equations are four laws that summarize how electric and magnetic fields are created and how they change. They connect electric charge, electric current, electric fields, magnetic fields, and light in one framework. These equations matter because they explain circuits, magnets, radio waves, optics, and much of modern technology.
They also show that light is an electromagnetic wave traveling at a predictable speed.
Key Facts
- Gauss's law for electricity: ∮ E · dA = Q_enclosed / ε0, electric charge creates electric field.
- Gauss's law for magnetism: ∮ B · dA = 0, there are no isolated magnetic monopoles in classical electromagnetism.
- Faraday's law: ∮ E · dl = -dΦB/dt, a changing magnetic flux creates a circulating electric field.
- Ampere-Maxwell law: ∮ B · dl = μ0 I_enclosed + μ0ε0 dΦE/dt, current and changing electric flux create magnetic field.
- Electromagnetic wave speed in vacuum: c = 1 / sqrt(μ0ε0) = 3.00 x 10^8 m/s.
- For a light wave in vacuum, E and B are perpendicular to each other and to the direction of travel, with E/B = c.
Vocabulary
- Electric field
- An electric field is a vector field that describes the force per unit positive charge at each point in space.
- Magnetic field
- A magnetic field is a vector field that describes magnetic forces on moving charges, currents, and magnetic materials.
- Flux
- Flux measures how much of a field passes through a surface, often found by multiplying field strength by area and the cosine of the angle.
- Displacement current
- Displacement current is the term μ0ε0 dΦE/dt in the Ampere-Maxwell law that allows a changing electric field to produce a magnetic field.
- Electromagnetic wave
- An electromagnetic wave is a traveling disturbance made of linked changing electric and magnetic fields.
Common Mistakes to Avoid
- Treating Gauss's law as saying the electric field is always Q/ε0A is wrong because that shortcut only works for highly symmetric cases where E is constant over the chosen surface.
- Forgetting the negative sign in Faraday's law is wrong because the minus sign represents Lenz's law, meaning the induced effect opposes the change in magnetic flux.
- Thinking magnetic field lines can start or end on magnetic charges is wrong in classical Maxwell theory because Gauss's law for magnetism says the net magnetic flux through any closed surface is zero.
- Assuming electric and magnetic fields in light point in the same direction is wrong because electromagnetic waves are transverse, so E, B, and the direction of travel are mutually perpendicular.
Practice Questions
- 1 A point charge of 2.0 μC is enclosed by a spherical surface. What is the total electric flux through the surface? Use ε0 = 8.85 x 10^-12 C^2/(N m^2).
- 2 In a vacuum electromagnetic wave, the electric field amplitude is 150 V/m. What is the magnetic field amplitude? Use c = 3.00 x 10^8 m/s and E/B = c.
- 3 Explain how Faraday's law and the Ampere-Maxwell law work together to allow a self-sustaining electromagnetic wave to travel through empty space.