Level 3 · Undergraduate core teaching kit · First- and second-year university
Electromagnetic fields and potentials
Use the learner record during the live investigation, then use the instructor guide to facilitate comparison, address misconceptions, and assess evidence-bounded reasoning.
Learner lab record
Gauge-equivalent potential reconstruction
Which plotted quantities can change under a gauge transformation while electric and magnetic observables remain unchanged?
Setup
Use the potential-and-gauge laboratory. Record a baseline scalar/vector potential, apply a declared gauge function, and compare potentials, fields, and any loop quantity separately.
Predict first
- 1. Predict which potential components change after adding a gradient.
- 2. Predict whether curl of the added gradient changes B.
| Variable | Role | Unit |
|---|---|---|
| Gauge-function amplitude | independent | model potential scale |
| Scalar and vector potentials | representation-dependent | V and field-potential units |
| Electric and magnetic fields | gauge-invariant dependent | V/m and T |
| Closed-loop phase/flux | gauge-invariant diagnostic | declared model unit |
Observation columns
Analyze
- 1. Which quantities changed without changing local fields?
- 2. What vector identity explains the magnetic result?
- 3. Why is gauge freedom not permission to invent a new force?
- 4. What combination would an experiment actually measure?
Conclusion frame
The gauge transformation changed ___ by ___ while E and B residuals remained ___; the physical prediction is unchanged because ___.
Instructor guide · 50–65 minutes
Teach the investigation, not the interface
Learning target: Learners distinguish representational gauge freedom from gauge-invariant fields, loops, and measured forces.
Prepare
- • Review gradient, curl, and ∇×∇χ=0.
- • Declare the laboratory's gauge transformation.
- • Prepare one claim that mistakes potential magnitude for local force.
Facilitation moves
- • Ask what the detector measures.
- • Compare potentials and fields in separate columns.
- • Require a named invariant before accepting a physical claim.
Accessibility and participation
- • Use numeric residuals alongside vector plots.
- • Describe gradient and curl operations verbally.
- • Ensure changed and invariant quantities use labels, not color alone.
Evidence of learning
- • A potential-versus-field comparison
- • A correct vector-identity explanation
- • One operationally measurable invariant
Misconception checks
Changing A mathematically creates a new magnetic field.
A pure gauge gradient changes the representation while its curl vanishes.
Potentials are therefore meaningless.
They are central computational objects, and gauge-invariant path or flux combinations can be observable.
Extension
Construct two gauges for the same uniform magnetic field and verify a closed-loop integral around the same contour.