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Mathematics & logicExamples and counterexamples

Learn what a claim covers, explain why it works, and recognise an allowed case that shows it needs changing.

MethodMathematics & logicCounting and zero

Count each object once, explain the final count, and distinguish quantity, position and identifying labels. Make zero’s role explicit.

DefinitionMathematics & logicNumber lines and order

Use number-line positions to compare whole numbers. Read the scale, recognise equality, and distinguish numerical distance from drawing size.

DefinitionMathematics & logicAddition and subtraction

Combine separate counts, remove a part, and find differences or missing parts. Connect objects, equations and number-line moves, then use tens and ones to regroup without changing value.

DefinitionMathematics & logicMultiplication and division

Count equal groups, distinguish sharing from making groups, and check with multiplication. Split calculations into parts; interpret zero and remainders in context.

DefinitionMathematics & logicFractions

Choose a whole, count equal fractional units, and connect sharing with division. Place fractions on the number line, including values beyond one, and explain what their numerator and denominator mean.

DefinitionMathematics & logic · Physics · MeasurementRatios and rates

Distinguish part-to-part ratios, fractions of a whole and rates with units. Scale quantities together, compare unit rates, and explain when a relationship is proportional. Check zero cases, starting amounts and whether a result fits its context.

DefinitionMathematics & logic · Physics · MeasurementQuantities and units

Describe a quantity as a numerical value with a unit, and distinguish its physical kind from its dimension. Build length, area, volume and rate units, check meaningful combinations, and separate exact unit relations from measurement uncertainty.

DefinitionMathematics & logic · Physics · MeasurementUnit conversions

Convert quantities with exact unit ratios, track cancellation through compound units, and apply the correct powers to area and volume. Distinguish temperature readings from differences, check dimensions and context, and retain measurement uncertainty when reporting.

MethodMathematics & logic · Physics · MeasurementEstimation and uncertainty

Distinguish exact model values, rounded approximations and estimates from measurements. Declare the rounding rule, preserve information through calculations and unit changes, and interpret resolution, precision, accuracy and the stated uncertainty type without turning a report’s digits into guaranteed bounds.

DefinitionMathematics & logicVariables and expressions

Use symbols to name numbers or quantities, describe their allowed values, and evaluate expressions by replacing every occurrence consistently. Preserve grouping, signs, fractions and units, and distinguish a calculated value from solving an equation or proving two rules equivalent.

DefinitionMathematics & logicEquality and equations

Read equality as a relationship, find the permitted values that make an equation true, and justify one-step solutions by reversible operations on both sides. Work exactly with negative numbers and fractions, recognize all-solution and no-solution cases, and check the original equation, units and context.

DefinitionMathematics & logic · PhysicsCoordinates and tables

Locate a point from an ordered pair, read named axes and their scales, and connect the same values in words, tables and plots. Keep signs, fractions and units exact, distinguish an axis point from a quadrant point, and separate selected observations from a stated rule and its domain before drawing connections.

DefinitionMathematics & logic · PhysicsSlope and straight lines

Read a straight line as a stated relationship, calculate its slope from consistent signed coordinate changes, and interpret its intersections with the axes. Keep horizontal, vertical and coincident-point cases distinct, attach units and domain conditions to context models, and use numerical scales rather than appearance alone to judge a line’s change.

DefinitionMathematics & logic · Physics · MeasurementReading graphs

Read a graph’s quantities, units, ticks and context before judging its visual impression. Calculate differences, reference-based percentages and interval rates from the represented values; distinguish legitimate zoom from distorted spacing; and keep samples, interpolation, missing information, uncertainty and causal explanations separate.

MethodPhysics · MeasurementObservables and measurement

Choose the physical quantity you want to investigate, specify the point, event or location involved, and explain how a reading relates to that quantity.

MethodPhysicsSystems and boundaries

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MethodPhysicsModels and assumptions

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MethodPhysicsHypotheses, laws and theories

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MethodMathematics & logic · PhysicsApproximations

Build a useful estimate, compare a correction with a stated reference scale, and check what is lost when you leave out a term.

ApproximationMathematics & logic · PhysicsSymmetry and invariance

Specify a transformation, identify what stays unchanged, and distinguish symmetry of an object from invariance of a physical model.

Mathematical statementPhysicsConservation laws

Use a defined system and a named quantity to track transfers, distinguish changing parts from conserved totals, and check the conditions of energy and momentum balances.

ModelPhysicsEmergence and levels of description

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ModelMathematics & logic · PhysicsVectors

Describe displacement with fixed-axis components, combine compatible vectors, and use small exact examples to introduce span, independence, basis and real vector spaces.

DefinitionPhysics · Information · MeasurementScientific records

Build an inspectable physics record, test and revise a bounded cart model, preserve processing history, and plan findable data with appropriate sharing conditions.

MethodPhysicsPosition, velocity and acceleration

Describe a point's motion on a chosen straight axis, then connect signed changes, instantaneous rates and one exact set of position, velocity and acceleration graphs.

DefinitionPhysicsNewton’s laws and forces

Choose one body, identify the forces acting on it, and use their vector sum to distinguish acceleration, balanced motion and mutual interactions.

ModelPhysicsWork, energy and power

Connect force and displacement to changes in motion, follow a spring’s stored energy, and distinguish an energy transfer from its rate.

ModelPhysics · MeasurementSpeed of light in vacuum

c denotes the speed of light in vacuum. Its numerical value is fixed exactly in SI units.

DefinitionPhysics · MeasurementPlanck constant

h denotes the Planck constant. Its numerical value is fixed exactly in SI units.

Definition
ModelSource checked

Maxwell equations in free space

Physics

Four field equations connect electric and magnetic fields with charge and current.

Key knowledge

The classical SI Maxwell equations in free-space form connect E and B to charge density ρ and current density J. They include Faraday induction and imply local charge conservation, ∂ρ/∂t + ∇·J = 0.

Values and equations

∇·E = ρ/ε₀; ∇·B = 0; ∇×E = −∂B/∂t; ∇×B = μ₀J + μ₀ε₀∂E/∂t; ∂ρ/∂t + ∇·J = 0

E
Electric field vector V m⁻¹
B
Magnetic flux-density vector T
ρ
Total electric charge per volume C m⁻³
J
Electric current-density vector A m⁻²
ε₀
Vacuum permittivity F m⁻¹
μ₀
Vacuum permeability H m⁻¹
t
Time s
∇
Spatial derivative operator; dot denotes divergence and cross denotes curl m⁻¹

Conditions and scope

  • These are the E/B equations with vacuum constants ε₀ and μ₀ and total charge/current sources; a source-free vacuum region has ρ = 0 and J = 0.
  • Use fields and sources smooth enough for the displayed differential operations. Singular sources or interfaces require integral or distributional treatment.
  • Macroscopic material D/H equations need constitutive relations; field solutions also need suitable initial/boundary data. No magnetic monopole source is included.
Sources · 3

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. Electromagnetic Fields and Energy (1989), Chapter 2: Maxwell’s differential laws in free spaceHermann A. Haus and James R. Melcher — MIT OpenCourseWareTextbook chapter · PDFocw.mit.eduRead source
  2. University Physics, volume 2: Maxwell's Equations and Electromagnetic WavesSamuel J. Ling, Jeff Sanny and William Moebs / OpenStax, Rice UniversityTextbook chapteropenstax.orgRead source
  3. Electromagnetism · Maxwell equationsDavid Tong · University of CambridgeCourse notes · PDFdamtp.cam.ac.ukRead source

Evidence and review

Haus and Melcher 2.3, 2.6 and Table 2.8.1: differential laws and continuity, rewritten with B = μ₀H in free space; OpenStax 16.1: four SI integral laws.

2026-10-10 · Four classical free-space equations and their charge-continuity consequence only; no numerical ε₀/μ₀, material law or full solution is certified.

Foundational review

2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.

Sources from other places

Scope or convention differs

The four differential E/B Maxwell equations and charge continuity agree for suitable charge/current densities.

The equations agree; the source’s old exact numerical value for vacuum permeability predates the present SI.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–22 of 22 relationships

Scientific relationships · 22 / 22

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Uses definition
    Relationship sources · 2

    E is the electric field in the classical free-space field equations; this is not a material D/H constitutive law.

  2. Uses definition
    Relationship sources · 1

    Current density is charge flow per area and obeys local continuity with charge density in this model.

  3. Related to
    Relationship sources · 1

    Linked statements

    • For linearly polarized light incident on an ideal linear analyzer, transmitted intensity is I = I₀ cos² θ, where θ separates the polarization and transmission axes.

    Linear optical electric-field direction and an ideal analyzer; no universal material response inferred.

  4. Related to
    Relationship sources · 1

    Linked statements

    • Locally, classical fields can be represented by B = ∇×A and E = −∇φ − ∂A/∂t. Compatible changes of the potentials can leave both fields unchanged.

    Local curl/gradient potentials represent the homogeneous Maxwell relations.

  5. Derived from
    Relationship sources · 1

    Linked statements

    • For a coil with fixed geometry and linear magnetic response, flux linkage NΦ = LI. With constant L, self-induced emf is −L dI/dt under the corresponding loop orientation.

    Linked magnetic flux and Faraday induction for a constant linear self-inductance.

  6. Derived from
    Relationship sources · 1

    Linked statements

    • In vacuum classical electrodynamics, u = ε₀E²/2 + B²/(2μ₀) is field energy density and S = E×B/μ₀ is its energy flux. Their fixed-volume balance includes J·E, the signed power transferred from the field to charges per unit volume (W m⁻³). A negative J·E means charges transfer energy to the field.

    Fixed-volume vacuum field-energy theorem with current work and outward energy flux.

  7. Related to
    Relationship sources · 2

    Linked statements

    • The Standard Model describes electromagnetic, weak and strong interactions with photon, W/Z and gluon fields respectively. Its Higgs boson is a spin-zero excitation. Gravity is outside this model.

    Electromagnetism is one sector; classical Maxwell fields describe a limit, not the complete Standard Model.

  8. Derived from
    Relationship sources · 2

    Linked statements

    • Electrostatic potential obeys ∇²φ=−ρ/ε₀ in vacuum; charge-free regions obey Laplace's equation. Specifying φ on the full boundary gives a unique Dirichlet solution. Compatible normal-derivative data give a Neumann solution up to an additive constant on a connected domain.

    Electrostatic Gauss law and curl-free potential give Poisson's equation with compatible boundary data.

  9. Derived from
    Relationship sources · 2

    Linked statements

    • A localized steady current distribution in vacuum produces B(r)=μ₀/(4π)∫J(r′)×(r−r′)/|r−r′|³d³r′. Charge conservation requires ∇·J=0; in this magnetostatic limit ∇×B=μ₀J and ∇·B=0.

    Steady divergence-free currents supply a magnetostatic field with the stated whole-space boundaries.

  10. Related to
    Relationship sources · 1

    Linked statements

    • Macroscopic electric polarization P gives bound volume charge ρ_b=−∇·P and surface charge σ_b=P·n at a material boundary. D=ε₀E+P satisfies ∇·D=ρ_free. In a local linear isotropic dielectric, P=ε₀χ_eE and D=εE, with ε=ε₀(1+χ_e).

    Bound-charge bookkeeping rewrites Gauss law in terms of macroscopic D and free charge.

  11. Related to
    Relationship sources · 2

    Linked statements

    • Magnetization M contributes bound volume current J_b=∇×M and surface current K_b=M×n. H=B/μ₀−M separates macroscopic magnetization from free currents, giving ∇×H=J_free+∂D/∂t. A local linear isotropic medium can use M=χ_mH and B=μH.

    Bound-current bookkeeping rewrites macroscopic magnetostatics in H and free currents.

  12. Derived from
    Relationship sources · 1

    Linked statements

    • In a homogeneous nonmagnetic isotropic dielectric, a monochromatic plane wave has k²=ε_r(ω)ω²/c² and B₀=k×E₀/ω. A real positive refractive index n=√ε_r gives phase velocity c/n; complex k describes attenuation and frequency dependence gives dispersion.

    Homogeneous linear constitutive response and harmonic Maxwell equations give the dispersion relation.

  13. Derived from
    Relationship sources · 1

    Linked statements

    • Vacuum fields E=−∇φ−∂_tA and B=∇×A are unchanged by A′=A+∇χ, φ′=φ−∂_tχ. In Lorenz gauge ∇·A+c⁻²∂_tφ=0, the potentials obey □φ=−ρ/ε₀ and □A=−μ₀J, with □=∇²−c⁻²∂_t² and c²=1/(ε₀μ₀).

    Vacuum potentials and a Lorenz constraint yield source wave equations while retaining initial/boundary gauge freedom.

  14. Derived from
    Relationship sources · 1

    Linked statements

    • For a homogeneous nonmagnetic ohmic medium with real ε and σ, a wave exp(ikz−iωt) satisfies k²=μ₀εω²+iμ₀σω. In the good-conductor regime σ≫εω at ω>0, k≈(1+i)/δ with δ=√[2/(μ₀σω)]. Field amplitude decays as exp(−z/δ); intensity decays twice as rapidly in the exponent.

    The homogeneous vacuum-permeability Ohmic constitutive relation gives the stated complex wave branch and good-conductor attenuation.

  15. Related to
    Relationship sources · 1

    Linked statements

    • In the local London model, ∇×j_s=−n_s q_s²B/m_s. Combined with quasistatic Maxwell equations this gives ∇²B=B/λ_L², where λ_L²=m_s/(μ₀n_s q_s²). A flat half-space screens weak fields exponentially. Persistent conductivity alone does not establish equilibrium flux expulsion.

    Ampère's law and zero magnetic divergence give the screening equation.

  16. Related to
    Relationship sources · 1

    Linked statements

    • For a gapped noninteracting two-dimensional band system, the occupied-band Berry curvature integrates to an integer C=(2π)⁻¹∫_BZTrF_xy d²k. Under the applicable insulating zero-temperature conditions, σ_xy=C e²/h. Interfaces carry net chiral edge modes set by the change in C; the invariant requires the separating gap to remain open.

    Hall conductivity is a measurable electromagnetic response under its transport setup.

  17. Related to
    Relationship sources · 3

    Linked statements

    • Yee FDTD staggers electric/magnetic fields in space and time to approximate Maxwell curl evolution. In a uniform isotropic nondispersive medium, the three-dimensional explicit scheme requires c_mΔt√(Δx⁻²+Δy⁻²+Δz⁻²)≤1. Stability alone does not guarantee correct phase, interfaces or absorption; finite PML and numerical dispersion need convergence tests.

    The method discretizes specified Maxwell initial/boundary dynamics.

  18. Related to
    Relationship sources · 1

    Linked statements

    • Ideal conducting-fluid motion evolves magnetic flux by induction and preserves an initially solenoidal field. Frozen flux applies only within that limit; reconnection requires nonideal effects or kinetic physics.

    Conceptual association supported by the recorded research scope; this relation does not assert a proven derivation or mandatory prerequisite.

  19. Related to
    Relationship sources · 1

    Linked statements

    • EM inversion links conductivity/permeability parameters to source, fields and receiver response. A primary-secondary calculation can resolve thin conducting casing and 3D reservoir variations on different meshes; derivatives must include model-dependent primary fields and source terms.

    Conceptual association supported by the recorded research scope; this relation does not assert a proven derivation or mandatory prerequisite.

  20. Related to
    Relationship sources · 2

    Linked statements

    • In the ideal macroscopic electrostatic conductor model, electric field is zero in the conducting material and each connected conductor is at a constant potential. Excess charge occupies its surfaces; a smooth surface in vacuum has E_out=σ_s n/ε₀ immediately outside. A cavity can contain a field if it contains charge, so field-free material does not guarantee an empty-field cavity under every condition.

    The two records describe related physical or metrological models or distinctions; the present claim does not require every hypothesis or result in the linked record.

  21. Related to
    Relationship sources · 2

    Linked statements

    • Magnetic flux through an oriented surface is Φ_B=∫B·n dA. For a fixed conducting loop, Faraday’s law gives induced electric emf ∮E·dl=−dΦ_B/dt with surface and boundary orientations related by the right-hand rule. N identical fixed turns linking the same flux give ε=−N dΦ_B/dt. The minus sign sets the circulation; any resulting current depends on the circuit response.

    The two records describe related physical or metrological models or distinctions; the present claim does not require every hypothesis or result in the linked record.

  22. Uses definition
    Relationship sources · 1

    Linked statements

    • The classical SI Maxwell equations in free-space form connect E and B to charge density ρ and current density J. They include Faraday induction and imply local charge conservation, ∂ρ/∂t + ∇·J = 0.

    The differential Maxwell equations use spatial divergence and curl. Smooth local fields in a specified Euclidean coordinate chart; interfaces may need integral/distributional forms.

Entry detailsPermanent link to this entry ↗
Record identity
physics-maxwell-equations
Revision
core.1
Structured reference record
Essential core · first collection · University core expansion