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Primitives and postulates — Examples

1 · Prerequisites

2 · Summary

Two worked comparisons of the primitive objects on the companion page em-primitives-and-postulates: one separating the source classes, one checking the SI dimensions of every displayed equation.

The first example contrasts a compactly supported smooth charge density $q f$ with the distributional point charge on a prescribed $C^{1}$ trajectory (Smooth and point-source constructions: distinct for nonzero charge, both zero at zero charge). The smooth density has total charge $q$ in both retained charge branches and belongs to the source class to which the smooth Maxwell branch refers; the point source is defined only by its spacetime pairings, satisfies distributional continuity, and is read at fixed time as the evaluation functional $q δ_{z (t)}$ . Assuming Countable Choice for the cited Lebesgue-volume, null-singleton and regular-distribution embedding suppliers, the example shows that for $q \neq 0$ no locally integrable density represents the point charge. The two constructions carry the same total charge but have distinct spatial distributions when $q \neq 0$ ; for $q = 0$ both give zero charge and current distributions, represented by smooth zero functions. The point-source pairings supply distributional continuity; neither locally integrable Maxwell branch supplies a nonzero singular-source Maxwell adoption. No field value at the particle and no measurement is asserted.

The second example verifies that every term of the four SI Maxwell equations and of the Lorentz coupling terms carries a matching unit (Dimensional check of every Maxwell and Lorentz term in SI units): $div E$ and $ρ / ε_{0}$ in $V m^{- 2}$ ; both source terms of $curl B$ and $curl B$ itself in $T m^{- 1}$ ; both sides of $curl E = - \partial_{t} B$ in $T s^{- 1}$ ; the adopted coupling terms $q Φ (U)$ and $m d U / d τ$ in newtons, the displayed coordinate terms $q E$ and $q (r^{'} \times B)$ in newtons, and $ρ E + J \times B$ in $N m^{- 3}$ ; and the energy density and flux $\frac{1}{2} (ε_{0} E^{2} + B^{2} / μ_{0})$ and $E \times B / μ_{0}$ in $J m^{- 3}$ and $W m^{- 2}$ . The computation is exact unit arithmetic: it is not a measurement, it proves no equation, and no numerical value of $μ_{0}$ is used or asserted.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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Smooth and point-source constructions: distinct for nonzero charge, both zero at zero charge

Physical scope

A comparison of two source classes of the classical framework in a named laboratory region: a compactly supported continuous (indeed smooth) charge density and the distributional point charge on a prescribed trajectory, which for nonzero charge is shown not to be represented by any locally integrable density; for zero charge both constructions give the zero distributions. The example is conditional on the stated definitions and on Countable Choice for the published embedding theorem's injectivity; it contains no measurement, no empirical claim and no assertion about fields at the singularity.

Example

This example is conditional on the definitions Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, The electric and magnetic fields as fixed-time spatial vector fields and the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches, and assumes Countable Choice ${A C}_{ω}$ both for the Lebesgue-volume source branch and for the injectivity of the regular-distribution embedding used below (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

(a) [Smooth source] Let $f \in C_{c}^{\infty} (R^{3})$ be nonnegative with Lebesgue normalization $\int_{R^{3}} f d λ_{3} = 1$ (the Riemann integral over any compact box containing $supp f$ has the same value by Riemann–Lebesgue comparison for distribution test integrands) (for instance any nonzero nonnegative smooth compactly supported bump divided by its finite nonzero integral), let $q \in R$ , and let $Ω \subseteq R^{3}$ be a nonempty open set containing $supp f$ . Then $ρ : = q f$ and $J : = 0$ is a smooth source field of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current: $ρ$ is compactly supported and of class $C^{\infty}$ , hence continuous, and for every bounded Lebesgue-measurable $V$ with compact closure inside $Ω$ and containing $supp f$ , the charge is $Q (V, t) = q \int f = q$ ; the same value holds for the retained compact Jordan/Riemann branch at every time $t$ , so the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches applies to this source.

(b) [Point source] Let $I \subseteq R$ be an open interval and $z : I \to R^{3}$ a $C^{1}$ trajectory whose graph map $t \mapsto (z (t), t)$ is proper into $Ω \times I$ (Point-support spacetime densities: distributions, continuity equation and endpoint fluxes; properness is automatic when $Ω \times I$ is the whole ambient domain or when $I = R$ ). For the same charge $q$ as in (a), the pairings $⟨ ρ, ψ ⟩ : = q \int_{I} ψ (z (t), t) d t$ and $⟨ J, ψ ⟩ : = q \int_{I} z^{'} (t) ψ (z (t), t) d t$ , for $ψ \in C_{c}^{\infty} (Ω \times I)$ , are the point-source distributions of Point-support spacetime densities: distributions, continuity equation and endpoint fluxes: they are distributions (Distribution, Test function space d of an open set) satisfying $\partial_{t} ρ + div J = 0$ in the distributional sense. At each fixed time $t$ this model assigns the charge $q \cdot 1_{z (t) \in V}$ to a Jordan-measurable $V \subseteq Ω$ , and its instantaneous spatial density is the evaluation functional $q δ_{z (t)}$ (Dirac delta and its derivatives), $φ \mapsto q φ (z (t))$ ; a source field of the smooth branch instead gives the integrals $φ \mapsto \int_{Ω} ρ (y, t) φ (y) d y$ of its continuous density.

(c) [The two classes differ] Fix $t \in I$ , put $z_{0} : = z (t) \in Ω$ and suppose $q \neq 0$ . Retain Countable Choice for the regular-distribution embedding and the Lebesgue-null singleton result cited below (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Then no locally integrable $ρ \in L_{l o c}^{1} (Ω)$ represents the point charge: there is no such $ρ$ whose regular distribution (Regular distribution from a locally integrable function) satisfies $⟨ ρ, φ ⟩ = q φ (z_{0})$ for every $φ \in C_{c}^{\infty} (Ω)$ . Indeed, if $ρ$ did, then on the open set $Ω ∖ {z_{0}}$ every test vanishes at $z_{0}$ , so the regular distribution of the restriction of $ρ$ is zero, and injectivity gives $ρ = 0$ almost everywhere there (Measure-null sets and almost-everywhere statements relative to a measure); since ${z_{0}}$ is Lebesgue-null (Every at most countable subset of $R^{n}$ is Lebesgue null; in particular $λ_{1} (Q) = 0$ ) and a union of two null sets is null (Finite and countable subadditivity of measures), $ρ = 0$ almost everywhere on $Ω$ . The regular distribution depends only on the almost-everywhere class, so $⟨ ρ, φ_{0} ⟩ = 0$ for a cutoff $φ_{0} \in C_{c}^{\infty} (Ω)$ with $φ_{0} = 1$ near $z_{0}$ (Test function cutoffs and euclidean localization), while the assumed identity gives $⟨ ρ, φ_{0} ⟩ = q φ_{0} (z_{0}) = q \neq 0$ : a contradiction. Hence the point source is not a source field of the smooth branch of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current (whose densities are continuous functions), and the smooth-branch equations are not applied to it pointwise; the point source enters the framework only through its distributional pairings.

(d) [Conclusion] The two source classes carry the same parameter $q$ and the same total charge $q$ (the smooth density integrates to $q$ ; the point model assigns charge $q$ to regions containing the particle). For $q \neq 0$ their spatial distributions are distinct and the point source has no locally integrable representative. For $q = 0$ both constructions give the zero charge and zero current distributions, represented by the smooth zero functions. The nonzero point source is used only through its pairings; neither branch of The microscopic vacuum Maxwell equations: smooth and weak branches supplies a singular-source Maxwell adoption. This example asserts no field value at $z (t)$ and contains no measurement or empirical claim.

Facts & Assumptions

Given: a nonnegative $f \in C_{c}^{\infty} (R^{3})$ with $\int_{R^{3}} f d λ_{3} = 1$ ; $q \in R$ ; a nonempty open $Ω \subseteq R^{3}$ with $supp f \subseteq Ω$ ; an open interval $I \subseteq R$ and a $C^{1}$ trajectory $z : I \to R^{3}$ whose graph map is proper into $Ω \times I$ ; a time $t \in I$ with $z_{0} : = z (t)$ ; and Countable Choice where the cited measure and embedding suppliers carry it.

[F1]

A smooth source field is a pair $(ρ, J)$ , continuous and $C^{1}$ wherever differentiated, with frame charge $Q (V, t) = \int_{V} ρ (y, t) d λ_{3}$ for bounded Lebesgue-measurable $V$ with compact closure in $Ω$ , and compact Jordan charge $Q_{R} (V, t) = \int_{V} ρ (y, t) d y$ ; the empty region has charge zero. (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current)

[F2]

The smooth Maxwell branch refers to jointly $C^{1}$ fields and continuous sources; the weak branch admits locally integrable fields and sources. Point or sheet sources that lack locally integrable representatives are excluded from both branches. (The microscopic vacuum Maxwell equations: smooth and weak branches)

[F3]

The point-source pairings $⟨ ρ, ψ ⟩ = q \int_{I} ψ (z (t), t) d t$ , $⟨ J, ψ ⟩ = q \int_{I} z^{'} (t) ψ (z (t), t) d t$ over an open interval $I$ with proper graph map define distributions satisfying $\partial_{t} ρ + div J = 0$ . (Point-support spacetime densities: distributions, continuity equation and endpoint fluxes)

[F4]

The Dirac distribution at $a$ is the evaluation functional $δ_{a} (φ) = φ (a)$ on the test space. (Dirac delta and its derivatives)

[F5]

The regular functional $u_{f} (φ) = \int_{Ω} f φ$ is complex-linear in $f$ and depends only on the almost-everywhere class of $f$ . (Regular distribution from a locally integrable function)

[F6]

Under Countable Choice, the map $f \mapsto u_{f}$ from $L_{l o c}^{1} (Ω)$ modulo almost-everywhere equality into distributions is injective. (Locally integrable functions embed in distributions)

[F7]

Under Countable Choice, every at most countable subset of $R^{n}$ is Lebesgue-null; in particular singletons are null. (Every at most countable subset of $R^{n}$ is Lebesgue null; in particular $λ_{1} (Q) = 0$ )

[F8]

A union of finitely many ( $μ$ -null) measurable sets is null, and almost-everywhere statements refer to a measurable null exceptional set. (Finite and countable subadditivity of measures, Measure-null sets and almost-everywhere statements relative to a measure)

[F9]

For compact $K \subseteq Ω$ with $Ω$ open there is $χ \in C_{c}^{\infty} (Ω)$ with $0 \leq χ \leq 1$ and $χ = 1$ on a neighborhood of $K$ . (Test function cutoffs and euclidean localization)

[F10]

On a compact Jordan region the bounded Borel zero extension of a continuous function is Riemann integrable and its Riemann and Lebesgue integrals agree; the Riemann integral over a Jordan set is defined by that zero extension. (Riemann–Lebesgue comparison for distribution test integrands, The Riemann integral of a bounded function over a bounded Jordan measurable set)

[F11]

The Lebesgue integral is linear on $L^{1}$ ; in particular $\int_{V} q g d λ_{3} = q \int_{V} g d λ_{3}$ for a real constant $q$ . (The Lebesgue integral is linear on $L^{1} (μ)$ , Integral over a measurable subset, Integrable real and complex functions, and their integrals)

Verification

technique · direct computations for the two source classes and a short contradiction for their difference

1.1F1F2F11

For the smooth source $ρ = q f$ , $J = 0$ : $ρ$ is $C^{\infty}$ with compact support and $J = 0$ is smooth, so $(ρ, J)$ is a smooth source field in the sense of [F1]; for every bounded Lebesgue-measurable $V$ with compact closure in $Ω$ containing $supp f$ one has $f 1_{V} = f$ pointwise, hence $Q (V, t) = \int_{V} q f d λ_{3} = q \int_{R^{3}} f d λ_{3} = q$ by [F11], and the same pair is therefore a source to which the smooth branch [F2] refers.

1.2F10F1

For the same $V$ required to be compact and Jordan measurable, the zero extension of $q f$ to a bounding rectangle is the bounded zero extension of a continuous function on a compact Jordan region, so it is Riemann integrable with $Q_{R} (V, t) = \int_{V} q f = q$ by [F10]; thus both charge branches of [F1] give the same value $q$ on this source, as the definition of $ρ = q f$ predicts.

1.3F3F4

For the point source, the pairings of [F3] applied to the given proper trajectory define distributions on $Ω \times I$ satisfying $\partial_{t} ρ + div J = 0$ ; its fixed-time reading is the evaluation functional $q δ_{z (t)}$ of [F4], in contrast with the fixed-time reading $φ \mapsto \int_{Ω} ρ (y, t) φ (y) d y$ of a smooth-branch density.

1.4F1F3F4F5

If $q = 0$ , then $q f = 0$ , the smooth current is zero, and both point-source pairings in [F3] vanish; likewise $q δ_{z (t)} = 0$ by [F4]. Hence both constructions give zero distributions represented by the smooth zero functions.

2.1step 1.3F5F6F7F8F9

Assume $q \neq 0$ and suppose a locally integrable $ρ \in L_{l o c}^{1} (Ω)$ satisfied $⟨ ρ, φ ⟩ = q φ (z_{0})$ for every $φ \in C_{c}^{\infty} (Ω)$ . Every test supported in the open set $Ω ∖ {z_{0}}$ vanishes at $z_{0}$ , so the regular distribution of $ρ ∣_{Ω ∖ {z_{0}}}$ is zero; by injectivity [F6] this gives $ρ = 0$ almost everywhere on $Ω ∖ {z_{0}}$ . The singleton ${z_{0}}$ is null by [F7], so their union is null by [F8] and $ρ = 0$ almost everywhere on $Ω$ ; since the regular distribution depends only on the almost-everywhere class [F5], $⟨ ρ, φ_{0} ⟩ = 0$ for the cutoff $φ_{0}$ of [F9] with $φ_{0} = 1$ on a neighborhood of the compact ${z_{0}} \subseteq Ω$ . But the assumed identity gives $⟨ ρ, φ_{0} ⟩ = q φ_{0} (z_{0}) = q \neq 0$ , a contradiction.

3.1step 1.1step 1.2step 1.3step 2.1step 1.4F1F2∎

Both constructions carry total charge $q$ by steps 1.1-1.3. For $q \neq 0$ , step 2.1 proves that the point charge has no locally integrable representative, so it differs from the smooth density and is covered by neither Maxwell branch of [F2]'s postulate. For $q = 0$ , step 1.4 gives the common zero distributions. Only the point-source pairings and their continuity identity have been established, with no singular-source Maxwell adoption, field value at $z (t)$ or measurement.

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Dimensional check of every Maxwell and Lorentz term in SI units

Physical scope

A dimensional consistency check of the adopted SI Maxwell equations and Lorentz coupling of the classical framework. It is exact unit arithmetic, not a measurement, not a proof of the equations and not a claim about the value of any constant; it inherits the framework's adopted equations only for the list of terms checked.

Example

This example is conditional on the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values, the definitions of divergence and curl in Divergence and curl of a $C^{1}$ vector field, the equations of The microscopic vacuum Maxwell equations: smooth and weak branches and the coupling of The Lorentz coupling: external force on a test charge and the continuum force and power densities; all equalities below are equalities of units, using $1 C = 1 A s$ , $1 V = 1 J C^{- 1}$ , $1 J = 1 N m$ , $1 W = 1 V A = 1 J s^{- 1}$ , $1 F = 1 C V^{- 1}$ and $1 T = 1 N A^{- 1} m^{- 1}$ .

(a) $div E$ has unit $V m^{- 2}$ and $ρ / ε_{0}$ has unit $C m^{- 3} / (F m^{- 1}) = C V m^{- 2} C^{- 1} = V m^{- 2}$ , so the two sides of $div E = ρ / ε_{0}$ agree.

(b) $curl B$ has unit $T m^{- 1}$ ; $μ_{0} J$ has unit $N A^{- 2} \cdot A m^{- 2} = N A^{- 1} m^{- 2} = T m^{- 1}$ ; $c^{- 2} \partial_{t} E$ has unit $m^{- 2} s^{2} \cdot V m^{- 1} s^{- 1} = V s m^{- 3} = N A^{- 1} m^{- 2} = T m^{- 1}$ , so both source terms of $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ have the unit of $curl B$ .

(c) $curl E$ has unit $V m^{- 2} = T s^{- 1}$ and $\partial_{t} B$ has unit $T s^{- 1}$ , so $curl E = - \partial_{t} B$ is homogeneous; $div B = 0$ is a homogeneous equation.

(d) In the coupling of The Lorentz coupling: external force on a test charge and the continuum force and power densities, the adopted right-hand side $q Φ (U)$ has unit $C \cdot T \cdot m s^{- 1} = C N A^{- 1} s^{- 1} = N$ (using $1 T = 1 N A^{- 1} m^{- 1}$ and $1 C = 1 A s$ ) and the left-hand side $m d U / d τ$ has unit $k g m s^{- 2} = N$ , so the two sides of $m d U / d τ = q Φ (U)$ agree; in the coordinate expression $q (E + r^{'} \times B)$ on the right of the postulate's displayed nonrelativistic acceleration equation (that whole equation is not adopted as exact), $q E$ has unit $C \cdot V m^{- 1} = J m^{- 1} = N$ and $q (r^{'} \times B)$ has unit $C m s^{- 1} T = C N A^{- 1} s^{- 1} = N$ , so its two terms add with the unit of a force; and $ρ E + J \times B$ has unit $C m^{- 3} V m^{- 1} = N m^{- 3}$ , the unit of a force density, while $J \cdot E$ has unit $A m^{- 2} \cdot V m^{- 1} = W m^{- 3}$ , the unit of a power density.

(e) The energy density $\frac{1}{2} (ε_{0} E^{2} + B^{2} / μ_{0})$ has unit $J m^{- 3}$ and the flux $E \times B / μ_{0}$ has unit $W m^{- 2}$ , consistent with $J m^{- 3} \cdot m s^{- 1} = W m^{- 2}$ .

(f) The check is exact arithmetic of the adopted units; it is not a measurement, it does not prove the equations, and no numerical value of $μ_{0}$ is used or asserted.

Facts & Assumptions

Given: the unit conventions of Framework-local SI units, the vacuum constants and the exact/measured status of their values, the component definitions of Divergence and curl of a $C^{1}$ vector field, and the adopted equations of The microscopic vacuum Maxwell equations: smooth and weak branches and The Lorentz coupling: external force on a test charge and the continuum force and power densities.

[F1]

The SI unit table: $c$ in $m s^{- 1}$ , $E$ in $V m^{- 1}$ , $B$ in T with $1 T = 1 N A^{- 1} m^{- 1}$ , $ρ$ in $C m^{- 3}$ , $J$ in $A m^{- 2}$ , force in N, $ε_{0} = (μ_{0} c^{2})^{- 1}$ in $F m^{- 1}$ with $μ_{0}$ in $N A^{- 2}$ , energy density in $J m^{- 3}$ and energy flux in $W m^{- 2}$ . (Framework-local SI units, the vacuum constants and the exact/measured status of their values)

[F2]

The divergence is $\sum_{i} \partial_{i} F_{i}$ and each curl component is a difference of two first partial derivatives, so divergence and curl divide the field unit by a length. (Divergence and curl of a $C^{1}$ vector field)

[F3]

The four Maxwell equations are $div E = ρ / ε_{0}$ , $div B = 0$ , $curl E = - \partial_{t} B$ and $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ . (The microscopic vacuum Maxwell equations: smooth and weak branches)

[F4]

The adopted coupling is $m d U / d τ = q Φ (U)$ , with the coordinate expression $q (E + r^{'} \times B)$ displayed in the nonrelativistic acceleration equation, which is not adopted as exact, and with the force density $ρ E + J \times B$ and the power density $J \cdot E$ . (The Lorentz coupling: external force on a test charge and the continuum force and power densities)

Verification

technique · direct unit arithmetic for each displayed term

1.1F1F2F3

Reading the units of $E$ , $ρ$ and $ε_{0}$ from [F1] and the component formula for the divergence from [F2]: $div E$ has unit $V m^{- 2}$ , while $ρ / ε_{0}$ has unit $C m^{- 3} / (F m^{- 1}) = C m^{- 3} \cdot m F^{- 1} = C V m^{- 2} C^{- 1} = V m^{- 2}$ , using $1 F = 1 C V^{- 1}$ ; the two sides of the first equation of [F3] therefore carry the same unit.

1.2F1F2F3

Again by [F1] and [F2], $curl B$ has unit $T m^{- 1}$ ; $μ_{0} J$ has unit $N A^{- 2} \cdot A m^{- 2} = N A^{- 1} m^{- 2} = T m^{- 1}$ ; and $c^{- 2} \partial_{t} E$ has unit $m^{- 2} s^{2} \cdot V m^{- 1} s^{- 1} = V s m^{- 3}$ , which equals $N A^{- 1} m^{- 2} = T m^{- 1}$ because $1 V = 1 J C^{- 1}$ , $1 J = 1 N m$ and $1 C = 1 A s$ ; hence all three terms of the fourth equation of [F3] carry the unit $T m^{- 1}$ .

1.3F1F2F3

By [F1] and [F2], $curl E$ has unit $V m^{- 2}$ and $\partial_{t} B$ has unit $T s^{- 1}$ ; since $V m^{- 2} = J C^{- 1} m^{- 2} = N m C^{- 1} m^{- 2} = N A^{- 1} s^{- 1} m^{- 1} = T s^{- 1}$ by the conversion identities, the third equation of [F3] is homogeneous, and the second equation of [F3] is homogeneous with $div B$ in $T m^{- 1}$ and a zero right-hand side of that same dimension.

1.4F1F4

By [F1] and [F4]: the adopted term $q Φ (U)$ has unit $C \cdot T \cdot m s^{- 1} = C N A^{- 1} s^{- 1} = N$ and $m d U / d τ$ has unit $k g m s^{- 2} = N$ ; in the displayed coordinate expression, $q E$ has unit $C \cdot V m^{- 1} = C \cdot J C^{- 1} m^{- 1} = N$ and $q (r^{'} \times B)$ has unit $C \cdot m s^{- 1} \cdot N A^{- 1} m^{- 1} = C N A^{- 1} s^{- 1} = N$ ; $ρ E$ has unit $C m^{- 3} \cdot V m^{- 1} = C V m^{- 4} = N m^{- 3}$ and $J \times B$ has unit $A m^{- 2} \cdot N A^{- 1} m^{- 1} = N m^{- 3}$ , so the force density $ρ E + J \times B$ has unit $N m^{- 3}$ ; and $J \cdot E$ has unit $A m^{- 2} \cdot V m^{- 1} = W m^{- 3}$ because $1 W = 1 V A$ .

1.5F1

By [F1]: $ε_{0} E^{2}$ has unit $F m^{- 1} \cdot V^{2} m^{- 2} = F V^{2} m^{- 3} = C V m^{- 3} = J m^{- 3}$ and $B^{2} / μ_{0}$ has unit $N^{2} A^{- 2} m^{- 2} \cdot A^{2} N^{- 1} = N m^{- 2} = J m^{- 3}$ , so the energy density $\frac{1}{2} (ε_{0} E^{2} + B^{2} / μ_{0})$ has unit $J m^{- 3}$ ; the flux $E \times B / μ_{0}$ has unit $V m^{- 1} \cdot N A^{- 1} m^{- 1} \cdot A^{2} N^{- 1} = V A m^{- 2} = W m^{- 2}$ .

2.1step 1.1step 1.2step 1.3step 1.4step 1.5F1F3F4∎

Every term listed in (a)-(e) of the example has been assigned the same unit as the term it is compared with, so each displayed Maxwell equation and coupling expression is dimensionally homogeneous in the adopted SI conventions; the computation used only the unit table and the component definitions, and no numerical value of $μ_{0}$ , no measurement and no claim that the equations hold was used.

Primitives and postulates — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Smooth and point-source constructions: distinct for nonzero charge, both zero at zero charge

Example

Facts & Assumptions

Verification

Dimensional check of every Maxwell and Lorentz term in SI units

Example

Facts & Assumptions

Verification

Sources