Maxwell's Equations Study Guide

1. Gauss's Law for Electricity

This equation relates the electric flux through a closed surface to the total charge enclosed by the surface:

∮_S E · dA = Q_enc / ε₀

• ∮_S represents the surface integral over a closed surface S
• E is the electric field
• dA is a differential area element
• Q_enc is the total enclosed charge
• ε₀ is the permittivity of free space

2. Gauss's Law for Magnetism

This equation states that the magnetic flux through a closed surface is always zero:

∮_S B · dA = 0

• B is the magnetic field
• dA is a differential area element

3. Faraday's Law of Electromagnetic Induction

This equation describes how a changing magnetic field induces an electric field:

∮_C E · dr = - dΦ_B / dt

• ∮_C represents the line integral around a closed path C
• E is the electric field
• dr is a differential path element
• dΦ_B / dt is the rate of change of magnetic flux

4. Ampère's Law with Maxwell's Addition

This equation relates the circulation of the magnetic field around a closed path to the sum of the current passing through the surface bounded by that path and the rate of change of electric flux through the surface:

∮_C B · dr = μ₀ (I_enc + ε₀ dΦ_E / dt)

• B is the magnetic field
• dr is a differential path element
• μ₀ is the permeability of free space
• I_enc is the total enclosed current
• dΦ_E / dt is the rate of change of electric flux