Electromagnetic fields and potentials
Derive Maxwell dynamics with vector calculus, potentials, gauges, waves, energy, and momentum.
Move from integral laws to differential equations, introduce scalar and vector potentials without treating gauge redundancy as a hidden force, and quantify cavity and radiation effects.
Before you begin
- • Calculus and vector fields
- • Level 2 electromagnetism
- • Ordinary differential equations
By the end, you can
- • Use integral and differential Maxwell equations.
- • Construct fields from scalar and vector potentials.
- • Explain gauge transformations and gauge-invariant observables.
- • Derive electromagnetic waves and energy-momentum flow.
Interactive model
Explore before calculating
Live laboratory
Potential, field, and gauge explorer
Move a sample point between two sources, then shift the scalar-potential zero. The potential value moves with the gauge offset while the observable field remains unchanged.
Potential φ: 0.400 normalized units
Field E: (0.384, 0.096)
|E|: 0.396
Gauge offset: 0.0
Adding a constant changes the potential label but not E = −∇φ. Gauge freedom changes description; measurable forces and gauge-invariant phase differences do not become arbitrary.
Level 3 · Undergraduate core teaching kit
Record the investigation. Teach the reasoning.
A learner-facing lab record and a course-specific instructor guide turn the live model into a repeatable classroom investigation.
Learner record
Gauge-equivalent potential reconstruction
Which plotted quantities can change under a gauge transformation while electric and magnetic observables remain unchanged?
Download learner recordInstructor guide
Teach for evidence, not button pushing
Learners distinguish representational gauge freedom from gauge-invariant fields, loops, and measured forces.
Download instructor guideLesson 1 of 3
Maxwell equations: local and global
How do sources inside a region determine flux and circulation on its boundary?
Gauss and Stokes theorems translate between differential laws and integral measurements. Charge sources electric divergence; magnetic divergence remains zero in standard electromagnetism.
Faraday induction and the Ampère–Maxwell law couple time-varying electric and magnetic fields.
Worked example
Use Gauss's law for a point charge with spherical symmetry.
- 1. Choose a sphere radius r.
- 2. E is radial and constant in magnitude on the sphere.
- 3. E(4πr²)=q/ε₀.
E=q/(4πε₀r²) r̂.
Try it
Integral-surface choice
Materials: Field diagrams and several candidate surfaces.
- 1. Identify symmetry.
- 2. Choose surfaces where field magnitude is constant or tangent.
- 3. Evaluate flux pieces.
- 4. Compare with enclosed source.
Notice: A clever boundary turns a difficult field integral into a simple algebraic relation.
Check your understanding: What does ∇·B=0 state?
Answer: Magnetic field has no observed local monopole source in standard Maxwell theory.
Magnetic flux lines form closed loops or extend through boundaries without beginning or ending locally.
Lesson 2 of 3
Potentials and gauge freedom
How can different potentials describe the same electromagnetic fields?
B=∇×A and E=−∇φ−∂A/∂t express fields using scalar and vector potentials. Adding a gauge-generated gradient and compensating scalar change leaves E and B unchanged.
Gauge freedom is descriptive redundancy, not permission to choose new observables. Quantum phases can depend on gauge-invariant loop information even where local fields vanish.
Worked example
Show A′=A+∇χ leaves B unchanged.
- 1. Compute B′=∇×A′.
- 2. Expand ∇×A+∇×∇χ.
- 3. Curl of a gradient is zero.
B′=B.
Try it
Gauge-equivalent potential maps
Materials: Notebook with a simple A field and chosen χ.
- 1. Compute A′=A+∇χ.
- 2. Take curls numerically.
- 3. Compare B and B′.
- 4. Identify which plotted features changed without physical field change.
Notice: Potential components can change dramatically while gauge-invariant fields remain identical.
Check your understanding: Does a nonzero vector potential at one point automatically imply a measurable local force?
Answer: No.
Local Lorentz force depends on E and B; quantum phase observables depend on gauge-invariant path or flux combinations.
Lesson 3 of 3
Waves, cavities, and stress-energy
How do fields transport energy and momentum through space and boundaries?
Source-free Maxwell equations yield waves with speed 1/√(μ₀ε₀). Boundary conditions select cavity modes and material response changes real-force predictions.
The Poynting vector gives energy flux, while the Maxwell stress tensor describes momentum transfer and pressure.
Worked example
Find time-averaged intensity of a plane wave with peak electric field E₀.
- 1. Magnetic amplitude is B₀=E₀/c.
- 2. Use S=(1/μ₀)E×B.
- 3. Average cos² over a cycle to 1/2.
⟨S⟩=½cε₀E₀².
Try it
Cavity mode solver
Materials: Notebook grid and rectangular-cavity dimensions.
- 1. Choose integer mode indices.
- 2. Compute allowed frequencies.
- 3. Plot field nodes.
- 4. Change one dimension and track spectral shifts.
Notice: Geometry selects modes, but the cavity's material and drive determine real energy and force.
Check your understanding: What quantity should be integrated to find electromagnetic power through a surface?
Answer: The Poynting vector dotted with the surface element.
∫S·dA gives net electromagnetic energy flow rate.
Formula-to-meaning deck
Read the equation in ordinary language.
∇·E=ρ/ε₀; ∇×B=μ₀J+μ₀ε₀∂E/∂t
Charge sources electric flux; current and changing electric field source magnetic circulation.
B=∇×A; E=−∇φ−∂A/∂t
Scalar and vector potentials generate observable fields.
S=(1/μ₀)E×B
Poynting vector gives electromagnetic energy flux.
Independent practice
Problem set
Work each problem before opening its hint and solution.
1. Find electric flux through a closed surface enclosing 3 nC.
Reveal hint
Use ΦE=q/ε₀.
Reveal solution
Approximately 339 N·m²/C.
2. Verify φ′=φ−∂χ/∂t with A′=A+∇χ leaves E unchanged.
Reveal hint
Substitute both primed potentials and cancel mixed derivatives.
Reveal solution
E′=−∇φ+∇∂tχ−∂tA−∂t∇χ=E.
3. A 2 mW beam is absorbed. Estimate force.
Reveal hint
Use F=P/c.
Reveal solution
About 6.7×10⁻¹² N.
Derivation studio
Build the result, line by line.
Keep the assumptions visible so the mathematics remains auditable.
Starting point
Electromagnetic wave equation
Source-free ∇×E=−∂B/∂t and ∇×B=μ₀ε₀∂E/∂t
- 1. Take curl of Faraday's law.
- 2. Use ∇×(∇×E)=∇(∇·E)−∇²E.
- 3. Set source-free ∇·E=0.
- 4. Substitute Ampère–Maxwell law.
∇²E−μ₀ε₀∂²E/∂t²=0
Coupled field laws contain waves with c=1/√(μ₀ε₀).
Starting point
Poynting theorem
Dot E into Ampère–Maxwell and B into Faraday
- 1. Subtract the two scalar products.
- 2. Use the vector identity for ∇·(E×B).
- 3. Group field-energy time derivatives.
- 4. Identify J·E as work on matter.
∂u/∂t+∇·S=−J·E
Local electromagnetic energy conservation separates field storage, flux, and transfer to charges.
Computational notebook
Turn the model into an experiment.
Rectangular cavity spectrum
How do geometry and mode indices shape a cavity's frequency density?
Inputs
- • Cavity side lengths
- • Mode-index limits
- • Optional material wave speed
Algorithm
- 1. Enumerate integer triplets.
- 2. Compute allowed frequencies.
- 3. Sort and histogram modes.
- 4. Perturb each dimension.
Evidence to produce
- • Frequency table and density histogram
- • Mode-shape slices
- • Sensitivity of resonances to geometry
Continue into the evidence
Source-linked next reading
Chapter 10: Longitudinal electrodynamics
Valid potential mathematics, Aharonov–Bohm physics, and the boundary against unsupported scalar claims.
Lecture 10: Pais patents
Apply Maxwell energy and field scales to proposed high-field devices.
Lecture 11: Eagleworks
Electromagnetic momentum and apparatus systematics in precision force tests.