The Spacetime Metric
Level 2 · Secondary physicsGrades 10–12About 9 hours

Electricity, magnetism, and Maxwell's picture

Connect charge, fields, circuits, induction, light, and electromagnetic momentum.

Build from Coulomb interactions to Maxwell's unified field picture, then use it to audit cavities, plasma devices, and propulsion experiments.

Established foundations

Before you begin

  • • Level 1 matter and fields
  • • Level 1 waves
  • • Mechanics and conservation laws

By the end, you can

  • • Calculate simple electric forces, fields, voltage, and current.
  • • Explain magnetic force and induction.
  • • Describe light as an electromagnetic wave.
  • • Track electromagnetic energy and momentum in an apparatus.

Interactive model

Explore before calculating

Transverse and longitudinal field components represented as distinct wave patterns.
Maxwell's equations specify which field patterns propagate in ordinary vacuum and how sources and boundaries shape them.

Live laboratory

Faraday–Lenz induction bench

Change a coil and its magnetic-flux interval. The induced voltage responds to flux rate, while Lenz's sign keeps the electrical output tied to work by the driver.

− emf+ emfzero

Flux change per turn: 2.00e-3 Wb

Induced emf: -1.600 V

Current: -0.160 A

Resistive power: 0.256 W

Induced direction: clockwise in the declared view. The opposing magnetic response means the agent changing the flux supplies the electrical energy delivered to the circuit.

Level 2 · Secondary physics 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

Faraday induction polarity map

How do flux-change rate and direction determine the magnitude and sign of an induced voltage?

Download learner record

Instructor guide

Teach for evidence, not button pushing

Learners connect induced-emf magnitude to flux-change rate and polarity to opposition of the change.

Download instructor guide
Open the complete print-friendly teaching kit →

Lesson 1 of 3

Charge, electric field, and voltage

How do local field and energy-per-charge descriptions fit together?

Electric field gives force per unit positive test charge. Voltage gives electric potential-energy difference per unit charge.

Field is a vector and voltage is a scalar. Both describe one electromagnetic system from complementary viewpoints.

electric fieldvoltagecurrentresistance

Worked example

A 2 μC charge in a 300 N/C field experiences what force?

  1. 1. Convert 2 μC to 2×10⁻⁶ C.
  2. 2. Use F = qE.
  3. 3. Multiply 2×10⁻⁶×300.

The force magnitude is 6×10⁻⁴ N, along the field for a positive charge.

Try it

Map equipotentials

Materials: A safe low-voltage field-mapping simulation.

  1. 1. Place two virtual electrodes.
  2. 2. Sample voltage on a grid.
  3. 3. Connect equal-voltage points.
  4. 4. Compare electric-field direction with the contours.

Notice: Electric-field arrows cross equipotential lines at right angles and point toward decreasing potential for positive test charge energy.

Check your understanding: Does high voltage necessarily mean high current?

Answer: No.

Current also depends on circuit resistance and connectivity; an open circuit can have voltage with negligible current.

Lesson 2 of 3

Magnetic force and induction

How can changing fields transfer energy without direct contact?

Magnetic fields act on moving charge and magnetic moments. The force direction is perpendicular to both velocity and field for a point charge.

A changing magnetic flux produces an electric field. Lenz's law sets the direction so induced effects oppose the flux change, preserving energy accounting.

magnetic fluxLorentz forceinductionLenz's law

Worked example

Why does a magnet slow when dropped through a conducting tube?

  1. 1. Changing flux induces circulating currents.
  2. 2. Those currents create a magnetic field opposing the change.
  3. 3. Mechanical energy becomes electrical resistance heating.

The braking is electromagnetic induction, not unexplained drag or lost energy.

Try it

Virtual induction lab

Materials: A coil-and-magnet simulation.

  1. 1. Move the magnet slowly through the coil.
  2. 2. Repeat faster.
  3. 3. Reverse the pole and motion.
  4. 4. Record voltage sign and peak size.

Notice: Faster flux change produces larger induced voltage; reversing the change reverses polarity.

Check your understanding: Why does Lenz's law oppose the change rather than reinforce it freely?

Answer: Reinforcement without extra input would violate energy conservation.

The reaction force makes the driver supply the electrical energy delivered to the circuit.

Lesson 3 of 3

Maxwell's equations and electromagnetic waves

How do changing electric and magnetic fields sustain light through space?

Maxwell's equations link charges to electric fields, currents to magnetic fields, changing magnetic fields to circulating electric fields, and changing electric fields to magnetic fields.

In source-free space those coupled changes form waves traveling at c. Their energy and momentum can exert radiation pressure.

Maxwell equationselectromagnetic wavePoynting vectorradiation pressure

Worked example

A 10 W beam is perfectly absorbed. What force scale can its momentum deliver?

  1. 1. For absorbed light, F = P/c.
  2. 2. Use c ≈ 3×10⁸ m/s.
  3. 3. Compute 10/(3×10⁸).

About 3.3×10⁻⁸ N—small but important in precision thrust experiments.

Try it

Cavity force budget

Materials: Paper, a cavity-power value, and a calculator.

  1. 1. Compute the maximum ordinary radiation-pressure scale P/c.
  2. 2. List cable forces, heating, outgassing, and magnetic coupling.
  3. 3. Rank expected sizes.
  4. 4. Design a reversal or null test.

Notice: Known electromagnetic effects create real tiny forces that must be bounded before claiming new propulsion.

Check your understanding: Can an electromagnetic wave carry momentum even though photons have zero rest mass?

Answer: Yes.

Relativistic energy and momentum are related; light transfers momentum when absorbed or reflected.

Formula-to-meaning deck

Read the equation in ordinary language.

F = qE; V = ΔU/q

Field sets force per charge; voltage sets energy difference per charge.

Units: N; V = J/C

F = q(v × B)

Magnetic force is perpendicular to charge motion and magnetic field.

Units: N

c = 1/√(μ₀ε₀); F_light ≈ P/c

Vacuum constants set wave speed; beam power sets radiation-pressure force scale.

Units: m/s; N

Independent practice

Problem set

Work each problem before opening its hint and solution.

  1. 1. A 12 V source drives 0.50 A. What power is delivered?

    Reveal hint

    Use P = VI.

    Reveal solution

    6.0 W.

  2. 2. Magnetic flux changes by 0.020 Wb in 0.10 s through one loop. Find average induced emf magnitude.

    Reveal hint

    Use |emf| = |ΔΦ/Δt|.

    Reveal solution

    0.20 V.

  3. 3. Estimate absorbed-light force from a 1 kW beam.

    Reveal hint

    Use P/c.

    Reveal solution

    About 3.3 μN.

Continue into the evidence

Source-linked next reading

Chapter 10: Longitudinal electrodynamics

Separates Maxwell/Whittaker mathematics from unsupported scalar-wave extrapolations.

Chapter 7: Electromagnetic metric engineering

Applies field and energy accounting to the Pais patent claims.

Lecture 10: Pais patent series

Documented mechanisms versus public experimental evidence.