GLOBAL KNOWLEDGE ARK

Reference library

Foundations

Explore concepts, laws, equations and methods, with clear explanations of their use and sources. Choose an area of study or find a specific entry.

More filters

Reference entries

Showing 181–210 of 1709 reference entries

InformationFinite graphs and trees

Graphs represent pairwise connections; trees impose acyclicity.

DefinitionInformationData-structure interfaces and operation costs

An interface defines operations; an implementation determines costs.

MethodInformationLoop invariants and total correctness

An invariant establishes correctness only with its proof obligations.

MethodInformationComparison sorting lower bound

Distinct-key comparisons need enough branches to distinguish orders.

Mathematical statementInformationBreadth-first and depth-first traversal

Graph searches visit reachable vertices using different orders.

MethodInformationShortest-path algorithm conditions

Edge weights determine which shortest-path method applies.

MethodInformationTuring machines and effective computation

Finite instructions can act on an unbounded writable tape.

DefinitionInformationRelations and tabular data

A relation is a set of tuples from declared domains.

DefinitionPhysicsCentre-of-mass dynamics

External force controls the mass-weighted centre's acceleration.

ModelPhysicsRigid-body inertia tensor

A tensor relates rotation to angular momentum.

ModelPhysicsEuler equations for a rigid body

Body-axis angular acceleration includes rotation of the axes.

ModelPhysics · Mathematics & logicEuler–Lagrange equations

Stationary action supplies coordinate equations of motion.

Mathematical statementPhysics · Mathematics & logicHamilton's canonical equations

Canonical coordinates and momenta evolve from the Hamiltonian.

Mathematical statementPhysicsFluid mass continuity

Density change balances mass flux through a region.

ModelPhysicsBernoulli's streamline relation

An ideal steady flow balances pressure, motion and height.

ModelPhysicsIncompressible Navier–Stokes equation

Fluid inertia balances pressure, viscosity and gravity.

ModelPhysicsReynolds number

Compare inertial and viscous scales of a flow.

ModelPhysicsDriven damped oscillator

A linear restoring force, damping and periodic drive determine response.

ModelPhysicsIdeal-string wave equation

A local differential equation determines small string disturbances.

ModelPhysics · Mathematics & logicWave superposition and interference

Linear disturbances add; their relative phase determines reinforcement.

ModelPhysicsFixed-end string modes

Boundary conditions select standing-wave frequencies.

ModelPhysicsSnell's refraction relation

Angles at an interface depend on refractive indices.

ModelPhysicsThin-lens image equation

A paraxial lens relates object and image distances.

ModelPhysicsSingle-slit diffraction minima

An aperture's finite width selects dark far-field directions.

ModelPhysicsLinear polarization and Malus's law

An ideal analyzer transmits the field component along its axis.

ModelPhysicsElectromagnetic potentials

Scalar and vector potentials encode electric and magnetic fields.

ModelPhysicsCapacitance

A linear two-conductor arrangement stores separated charge.

DefinitionPhysicsSelf-inductance

Changing current changes linked magnetic flux and induced emf.

ModelPhysicsElectromagnetic energy balance

The rate of field-energy change balances outward energy flow and power transferred to charges.

ModelPhysicsKirchhoff circuit balances

Lumped circuit models balance node currents and loop voltages.

Model
ModelSource checked

Electromagnetic energy balance

Physics

The rate of field-energy change balances outward energy flow and power transferred to charges.

Key knowledge

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.

Values and equations

d/dt ∫_V u dV + ∮_∂V S·n dA = −∫_V J·E dV; u = ε₀E²/2 + B²/(2μ₀); S = E×B/μ₀

u
Field energy density J m⁻³
S
Energy flux W m⁻²
J
Charge current density (excluding displacement current) A m⁻²
E
Electric field V m⁻¹
B
Magnetic flux density T
ε₀, μ₀
Vacuum permittivity and permeability F m⁻¹, H m⁻¹
n
Outward unit normal
dV, dA
Volume and area elements m³, m²
t
Time s
J·E
Signed power transferred from the field to charges per unit volume W m⁻³

Conditions and scope

  • Use a fixed control volume with outward normal and fields regular enough for the integrals.
  • J includes currents in that volume; material polarization and magnetization require additional energy accounting.

Selected sections only. Arbitrary material energy formulas, quantum fields, irreversible-heat assumptions and universal positive J·E are not adopted; no whole-source revalidation or human certification.

Sources · 2

References

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

  1. Electromagnetics and Applications 6.013, Lecture 4: Poynting theoremMIT OpenCourseWareCourse notes · PDFocw.mit.eduRead source
  2. Electromagnetic Energy Conservation (2014): equations 100–110Richard Fitzpatrick / University of Texas at AustinCourse notesfarside.ph.utexas.eduRead source

Evidence and review

Lecture 4, Poynting theorem slide: integral energy, boundary flux and E·J work.

Fresh 2026-10-11 selected source inspection: MIT Lecture 4, L4-2–L4-4 / PDF pages 2–4 distinguishes u [J/m³], S [W/m²], E·J [W/m³] and integrated power [W]. Fitzpatrick equations 100 and 108–110 and their prose give charge-current/field energy exchange and the signed power-density interpretation. Original fixed-volume balance, outward normal, two conditions and formula are retained.

2026-10-10 · Vacuum fixed-volume balance only; dielectric energy formulas are not generalized to arbitrary media.

2026-10-11 · Author/editor checked the power-versus-energy density dimensions and selected MIT/UT source bodies. The summary, J·E signed power interpretation and J charge-current label are clarified; original equation and conditions remain exact. The new independent record/coverage adoption review is pending. Historical review 2026-10-10 remains retained with its original scope.

Related reference entries

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

Showing 1–5 of 5 relationships

Scientific relationships · 5 / 5

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

  1. 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.

  2. Related to
    Relationship sources · 1

    Linked statements

    • A nonrelativistic accelerated point charge radiates total power P=q²a²/(6πε₀c³). Its far radiation field decreases as 1/r and has the transverse sin²θ angular power pattern relative to the acceleration.

    The nonrelativistic acceleration field's far-zone Poynting flux gives the displayed total radiated power.

  3. Derived from
    Relationship sources · 2

    Linked statements

    • A localized nonrelativistic electric dipole radiates E_rad≈n×(n×p̈(t−r/c))/(4πε₀c²r) and B_rad=n×E_rad/c. For p(t)=p₀ cosωt along a fixed axis, the cycle-averaged angular power is ω⁴p₀²sin²θ/(32π²ε₀c³) and the total average power is ω⁴p₀²/(12πε₀c³).

    Vacuum E/B radiation fields give the angular outgoing Poynting power and its full-solid-angle integral.

  4. Related to
    Relationship sources · 1

    Linked statements

    • With passive sign convention, current entering an element’s higher-potential terminal gives absorbed instantaneous power p=vi and energy ∫pdt. For a positive constant resistance, p=i²R=v²/R≥0: electrical energy becomes internal energy and other dissipation channels. AC average power is the average of vi, rather than generally the product of rms voltage and rms current.

    This relation concerns the stated electrical power and joule heating claim and its conditions; it does not assert an unqualified prerequisite or an additional theorem.

  5. Related to
    Relationship sources · 1

    Linked statements

    • For a fixed linear capacitance C>0 charged from zero, stored energy is U=Q²/(2C)=CV²/2=QV/2. The factor one-half comes from integrating the rising voltage during charging, not multiplying the final voltage by every charge increment. An ideal vacuum parallel-plate capacitor of area A and gap d has C≈ε₀A/d when edge fields are negligible.

    This relation concerns the stated energy stored in a linear capacitor claim and its conditions; it does not assert an unqualified prerequisite or an additional theorem.

Entry detailsPermanent link to this entry ↗
Record identity
physics-poynting-balance
Revision
university-reference.1.poynting-precision-20261011.1
Structured reference record
University reference · University core expansion