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Electromagnetism Maxwell Equations Differential Form cheat sheet - grade college

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Physics Grade college

Electromagnetism Maxwell Equations Differential Form Cheat Sheet

A printable reference covering Gauss’s laws, Faraday’s law, Ampere-Maxwell law, charge continuity, and constitutive relations for college physics.

TopicsElectromagnetismElectricityMagnetismDerivativesVectorsPhysicsStudy Guide
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Maxwell’s equations in differential form describe how electric and magnetic fields behave at each point in space and time. This cheat sheet helps college students connect local field equations to charge density, current density, and material properties. It is especially useful for solving symmetry problems, interpreting wave propagation, and checking signs in vector calculus forms.

The four core equations are ∇⋅E=ρε0\nabla \cdot \mathbf{E}=\frac{\rho}{\varepsilon_0}, ∇⋅B=0\nabla \cdot \mathbf{B}=0, ∇×E=−∂B∂t\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}, and ∇×B=μ0J+μ0ε0∂E∂t\nabla \times \mathbf{B}=\mu_0\mathbf{J}+\mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t}. Together they show that charges create electric field divergence, magnetic monopoles are absent, changing magnetic fields create circulating electric fields, and currents plus changing electric fields create circulating magnetic fields. In matter, the fields are often written using D\mathbf{D} and H\mathbf{H} with constitutive relations such as D=εE\mathbf{D}=\varepsilon\mathbf{E} and B=μH\mathbf{B}=\mu\mathbf{H}.

Key Facts

  • Gauss’s law for electricity in differential form is ∇⋅E=ρε0\nabla \cdot \mathbf{E}=\frac{\rho}{\varepsilon_0}, so electric charge density is the local source of electric field divergence.
  • Gauss’s law for magnetism is ∇⋅B=0\nabla \cdot \mathbf{B}=0, meaning magnetic field lines have no beginning or end in classical electromagnetism.
  • Faraday’s law is ∇×E=−∂B∂t\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}, so a time-changing magnetic field produces a circulating electric field.
  • The Ampere-Maxwell law is ∇×B=μ0J+μ0ε0∂E∂t\nabla \times \mathbf{B}=\mu_0\mathbf{J}+\mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t}, where μ0ε0∂E∂t\mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t} is the displacement current term.
  • Charge conservation follows from Maxwell’s equations as ∇⋅J+∂ρ∂t=0\nabla \cdot \mathbf{J}+\frac{\partial \rho}{\partial t}=0.
  • In vacuum, electromagnetic waves satisfy c=1μ0ε0c=\frac{1}{\sqrt{\mu_0\varepsilon_0}} and travel at the speed of light.
  • For a linear isotropic medium, the common constitutive relations are D=εE\mathbf{D}=\varepsilon\mathbf{E}, B=μH\mathbf{B}=\mu\mathbf{H}, and J=σE\mathbf{J}=\sigma\mathbf{E}.
  • In macroscopic form, Maxwell’s equations are ∇⋅D=ρf\nabla \cdot \mathbf{D}=\rho_f, ∇⋅B=0\nabla \cdot \mathbf{B}=0, ∇×E=−∂B∂t\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}, and ∇×H=Jf+∂D∂t\nabla \times \mathbf{H}=\mathbf{J}_f+\frac{\partial \mathbf{D}}{\partial t}.

Vocabulary

Divergence
Divergence, written ∇⋅F\nabla \cdot \mathbf{F}, measures the local source or sink strength of a vector field.
Curl
Curl, written ∇×F\nabla \times \mathbf{F}, measures the local circulation or rotation of a vector field.
Charge density
Charge density ρ\rho is electric charge per unit volume, usually measured in C/m3\mathrm{C}/\mathrm{m}^3.
Current density
Current density J\mathbf{J} is electric current per unit area, usually measured in A/m2\mathrm{A}/\mathrm{m}^2.
Displacement current
Displacement current is the term ε0∂E∂t\varepsilon_0\frac{\partial \mathbf{E}}{\partial t} that allows changing electric fields to create magnetic fields.
Constitutive relation
A constitutive relation connects field quantities inside a material, such as D=εE\mathbf{D}=\varepsilon\mathbf{E} or B=μH\mathbf{B}=\mu\mathbf{H}.

Common Mistakes to Avoid

  • Dropping the displacement current term in ∇×B=μ0J+μ0ε0∂E∂t\nabla \times \mathbf{B}=\mu_0\mathbf{J}+\mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t} is wrong because it breaks charge conservation and fails for changing electric fields.
  • Using ∇⋅B=ρm\nabla \cdot \mathbf{B}=\rho_m is wrong in standard classical electromagnetism because magnetic monopoles are not included, so the correct equation is ∇⋅B=0\nabla \cdot \mathbf{B}=0.
  • Reversing the sign in Faraday’s law is wrong because ∇×E=−∂B∂t\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} includes Lenz’s law, which sets the induced field direction.
  • Confusing E\mathbf{E} with D\mathbf{D} or B\mathbf{B} with H\mathbf{H} is wrong because microscopic vacuum equations and macroscopic material equations use different source terms and constants.
  • Treating differential equations as global statements is wrong because ∇⋅E=ρε0\nabla \cdot \mathbf{E}=\frac{\rho}{\varepsilon_0} describes a local point-by-point relationship, not directly a total flux without integration.

Practice Questions

  1. 1 At a point in vacuum, the charge density is ρ=3.54×10−9 C/m3\rho=3.54\times 10^{-9}\,\mathrm{C}/\mathrm{m}^3. Find ∇⋅E\nabla \cdot \mathbf{E} using ε0=8.85×10−12 F/m\varepsilon_0=8.85\times 10^{-12}\,\mathrm{F}/\mathrm{m}.
  2. 2 In a region with J=0\mathbf{J}=0, the electric field changes at a rate ∂E∂t=2.0×106 V/(m⋅s) x^\frac{\partial \mathbf{E}}{\partial t}=2.0\times 10^{6}\,\mathrm{V}/(\mathrm{m}\cdot\mathrm{s})\,\hat{\mathbf{x}}. Find the displacement contribution μ0ε0∂E∂t\mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t} to ∇×B\nabla \times \mathbf{B}.
  3. 3 For a material with ε=4ε0\varepsilon=4\varepsilon_0 and electric field magnitude E=150 V/mE=150\,\mathrm{V}/\mathrm{m}, calculate the magnitude of D\mathbf{D} using D=εE\mathbf{D}=\varepsilon\mathbf{E}.
  4. 4 Explain why the equation ∇⋅B=0\nabla \cdot \mathbf{B}=0 implies that magnetic field lines form closed loops or extend without endpoints.