Chapter 18
Gauss’s Law (Electric) as Field Integration
Field and Energy Systems
Author: Mark Anthony Higgins
Revised Date: 23 February 2026
Chapter 18
Flow Algebra
flow "GaussLaw_Electric_FieldIntegration"
# ---------- STATES ----------
state:
# field + sources
charge.density.ρ: fn(vec3)->float # volume charge density ρ(r) [C/m^3]
charge.q_enc: float # total enclosed charge [C]
field.E: fn(vec3)->vec3 # electric field E(r) [V/m]
potential.Φ: fn(vec3)->float # electrostatic potential Φ(r) [V]
# geometry for integration
surface.S: surface # closed surface S with outward normal n̂
volume.V: volume # volume V bounded by S
normal.nhat: fn(vec3)->vec3 # unit normal on S
area.element.dA: fn(vec3)->float # differential area element on S
vol.element.dV: fn(vec3)->float # differential volume element in V
# flux / divergence
flux.ΦE: float # total electric flux through S
divergence.∇·E: fn(vec3)->float # divergence field
permittivity.ε0: float # vacuum permittivity [F/m]
# symmetry helpers (optional)
symmetry.type: "spherical"|"cylindrical"|"planar"|"none"
gaussian.radius.R: float # sphere radius (spherical)
gaussian.length.L: float # cylinder length (cylindrical)
gaussian.area.A: float # area (planar sheet)
# coherence view
coherence.current.Jcohμ: vec[float,4] # [ρE, S] coherence-energy current analogue
boundary.integrity: float # 0..1 (1 = perfect stationary coherence)
time.t: float
# ---------- PARAMETERS ----------
params:
ε0: 8.8541878128e-12
k_e: 8.9875517923e9
discretise.Ns: int # surface sample count
discretise.Nv: int # volume sample count
dt: float
# ---------- INVARIANTS ----------
invariant:
# Gauss (integral form)
- flux.ΦE = ∬_S field.E(r) · normal.nhat(r) dA
# Gauss (differential form)
- divergence.∇·E(r) = charge.density.ρ(r) / params.ε0
# Divergence theorem consistency
- ∬_S E·dA = ∭_V (∇·E) dV
# Enclosed charge
- charge.q_enc = ∭_V charge.density.ρ(r) dV
- flux.ΦE = charge.q_enc / params.ε0
# Superposition
- field.E = Σ_i field.E_i ; charge.density.ρ = Σ_i ρ_i
# Coherence mapping (HST)
- ∂μ Jcoh^μ = 0 ↔ ∇·E = ρ/ε0 (field coherence continuity)
# ---------- TRANSFORMS ----------
transform:
initialise_problem:
do:
permittivity.ε0 = params.ε0
boundary.integrity = 0.995
time.t = 0
set_symmetry_surface:
do:
match symmetry.type:
- "spherical":
# choose sphere of radius R centred on symmetry centre
surface.S = sphere(R = gaussian.radius.R)
volume.V = ball(R = gaussian.radius.R)
- "cylindrical":
surface.S = closed_cylinder(R = gaussian.radius.R, L = gaussian.length.L)
volume.V = cylinder(R = gaussian.radius.R, L = gaussian.length.L)
- "planar":
surface.S = pillbox(area = gaussian.area.A, height = h_small())
volume.V = pillbox_volume(area = gaussian.area.A, height = h_small())
- "none":
surface.S = user_defined_surface()
volume.V = user_defined_volume()
compute_flux_integral:
do:
# numeric surface integration (fallback when no symmetry shortcut)
flux.ΦE = 0
for k in 1..params.discretise.Ns:
r_k = sample_on(surface.S, k)
dA_k = area.element.dA(r_k)
n_k = normal.nhat(r_k)
flux.ΦE += dot(field.E(r_k), n_k) * dA_k
compute_volume_charge:
do:
charge.q_enc = 0
for j in 1..params.discretise.Nv:
r_j = sample_in(volume.V, j)
dV_j = vol.element.dV(r_j)
charge.q_enc += charge.density.ρ(r_j) * dV_j
verify_gauss_equivalence:
do:
# differential-side volume integral
vol_div = 0
for j in 1..params.discretise.Nv:
r_j = sample_in(volume.V, j)
dV_j = vol.element.dV(r_j)
vol_div += divergence.∇·E(r_j) * dV_j
assert approx_equal(flux.ΦE, vol_div, rel=1e-3)
assert approx_equal(flux.ΦE, charge.q_enc / params.ε0, rel=1e-3)
symmetry_shortcuts:
do:
match symmetry.type:
- "spherical":
# E is radial and constant over sphere: ΦE = E( R ) * 4πR^2
E_mag = magnitude(field.E([gaussian.radius.R,0,0]))
flux.ΦE = E_mag * 4π * gaussian.radius.R^2
- "cylindrical":
# For infinite line charge λ: ΦE_side = E * (2πR L)
# (end caps contribute ~0 by symmetry)
E_mag = magnitude(field.E([gaussian.radius.R,0,0]))
flux.ΦE = E_mag * (2π * gaussian.radius.R * gaussian.length.L)
- "planar":
# For infinite sheet σ: E · n̂ = σ/(2ε0) per side; pillbox captures 2 faces
# ΦE = E_top A + E_bottom A = (σ/2ε0)A + (σ/2ε0)A = σA/ε0
σ = surface_charge_density(volume.V) # effective via ρ→σ
flux.ΦE = σ * gaussian.area.A / params.ε0
update_coherence_view:
do:
# map energy density into a coherence current picture (HST)
ρE = 0.5 * params.ε0 * |field.E(sample_center(volume.V))|^2
S_vec = [0,0,0] # electrostatics → no time-avg Poynting flow
coherence.current.Jcohμ = [ρE, S_vec]
boundary.integrity = min(1.0, boundary.integrity + 1e-4)
tick:
do:
time.t += params.dt
# ---------- MONITORS ----------
monitor:
flux_vs_charge:
observe: [flux.ΦE, charge.q_enc, permittivity.ε0]
report: "Checks Φ_E = q_enc / ε0 across chosen Gaussian surface."
surface_vs_volume:
observe: [flux.ΦE]
report: "Verifies ∬_S E·dA = ∭_V (∇·E) dV by divergence theorem."
symmetry_health:
observe: [symmetry.type]
report: "Symmetric surfaces (sphere/cylinder/pillbox) minimise computation and expose field constants on S."
coherence_consistency:
observe: [boundary.integrity, coherence.current.Jcohμ]
report: "Electrostatic fields are stationary coherence: ∂μ Jcoh^μ = 0 ↔ ∇·E = ρ/ε0."
# ---------- NOTES (coherence view) ----------
note:
- **Gauss (integral):** ∬_S E·dA = q_enc/ε0. **Differential:** ∇·E = ρ/ε0. They are equivalent via the divergence theorem.
- Choose a **Gaussian surface aligned with symmetry** so E is constant on S (sphere → point charge, cylinder → line charge, pillbox → sheet).
- Superposition holds: both ρ and E add linearly; Φ_E and q_enc scale accordingly.
- In HST, Gauss’s law is the **continuity of field coherence**: ∂μ J_coh^μ = 0 maps to ∇·E = ρ/ε0 in electrostatics.
- Field energy density ρ_E = ½ ε0 |E|² acts as curvature tension; electrostatics stores tension (no net radiative flux).
Worked Example — Point Charge (Spherical symmetry)
preset "PointCharge_Sphere"
given:
q := 1.0e-6 # C
ε0 := 8.8541878128e-12 # F/m
R := 0.25 # m (Gaussian sphere radius)
set:
symmetry.type := "spherical"
gaussian.radius.R := R
charge.density.ρ(r) := q * δ³(r) # idealised point charge at the origin
field.E(r) := (1/(4π ε0)) * q * r̂ / |r|^2 # Coulomb field
compute:
# Symmetry shortcut: E is constant over the sphere, radial.
E_mag := (1/(4π ε0)) * q / R^2
flux.ΦE := E_mag * (4π R^2) # = q/ε0
check:
q_enc := ∬∬_V ρ dV = q
expect:
flux.ΦE ≈ q_enc / ε0
flux.ΦE = ( (1/(4π ε0)) * q / R^2 ) * (4π R^2 ) = q/ε0 # ✓ independent of R
\[ \text{Result: } \Phi_{E} = \frac{q}{\varepsilon_{0}}\,. \] \[ \text{The surface integral equals the enclosed charge over } \varepsilon_{0}, \text{ independent of sphere size.} \]
Worked Example — Infinite Line Charge (Cylindrical symmetry)
preset "LineCharge_Cylinder"
given:
λ := 2.0e-6 # C/m (line charge density)
ε0 := 8.8541878128e-12
R := 0.10 # m (Gaussian cylinder radius)
L := 0.50 # m (Gaussian cylinder length)
set:
symmetry.type := "cylindrical"
gaussian.radius.R := R
gaussian.length.L := L
charge.density.ρ(r) := λ δ²(r⊥) # line along z-axis
field.E(r) := (λ/(2π ε0 r)) r̂ # radial, ~1/r
compute:
# Flux through curved side only; end caps contribute ~0 by symmetry.
E_mag := λ/(2π ε0 R)
flux.ΦE := E_mag * (2π R L) # = λL/ε0
check:
q_enc := λ L
expect:
flux.ΦE ≈ q_enc / ε0 # ✓ independent of R and matches Gauss
q_enc = λ L
check:
assert approx_equal(flux.ΦE, q_enc/ε0)
numeric:
E_mag ≈ (2e-6)/(2π*8.854e-12*0.10) ≈ 3.60e5 V/m
ΦE = (λL)/ε0 ≈ (2e-6)/(8.854e-12) ≈ 2.26e5 N·m^2/C
\[ \text{Result: } \Phi_{E} = \frac{\lambda\,L}{\varepsilon_{0}}\,. \]
Worked Example — Infinite Sheet (Planar symmetry)
preset "SheetCharge_Pillbox"
given:
σ := 5.0e-6 # C/m^2 (surface charge density)
ε0 := 8.8541878128e-12
A := 0.40 # m^2 (pillbox face area)
set:
symmetry.type := "planar"
gaussian.area.A := A
volume.V := thin_pillbox(area=A, height=h≪1)
field.E_above := σ/(2 ε0) n̂
field.E_below := -σ/(2 ε0) n̂ # opposite side
compute:
# Flux = E_top·A + E_bottom·A
flux.ΦE := (σ/(2 ε0))A + (σ/(2 ε0))A = σA/ε0
check:
q_enc := σA
expect:
flux.ΦE ≈ q_enc / ε0 # ✓ pillbox captures both faces
\[ \text{Result: } \Phi_{E} = \frac{\sigma\,A}{\varepsilon_{0}}\,. \] \[ \text{The pillbox neatly counts both faces, giving the familiar } E = \frac{\sigma}{2\varepsilon_{0}} \text{ per side.} \]
Common Pitfalls (and quick fixes)
Forgetting end caps / wrong surface choice
\( \Phi_{E} = E_{\text{top}}A + E_{\text{bottom}}A = \dfrac{\sigma}{2\varepsilon_{0}}A + \dfrac{\sigma}{2\varepsilon_{0}}A = \dfrac{\sigma A}{\varepsilon_{0}} \)
Mixing up the sign/placement in Gauss & Poisson
\[ \iint_{S} \mathbf{E} \!\cdot\! d\mathbf{A} \;=\; \iiint_{V} (\nabla \!\cdot\! \mathbf{E})\, dV\,. \]
Forgetting superposition
For multiple sources:
\[ \Phi_{\mathrm{total}} = \sum_{i} \Phi_{i}, \qquad \mathbf{E}_{\mathrm{total}} = \sum_{i} \mathbf{E}_{i}, \qquad q_{\mathrm{enc}} = \sum_{i} q_{i}\,. \]
\[ \Phi_{E} = \frac{q_{\mathrm{enc}}}{\varepsilon_{0}}\,. \]
Changing \( R \) changes \( \mathbf{E} \) and area oppositely; the product stays fixed.
Delta-function sources mishandled
Units drift
Forgetting field direction
Confusing enclosed vs. external charge
Mixing Gaussian & SI constants
Gauss’s Law (Electric) — Field Integration
Flow Summary
Gauss’s Law relates the flux of the electric field through any closed surface to the charge enclosed within that surface.
In the HST framework, this law describes the continuity of coherence flow across a curvature boundary — the same conservation law that governs all stable fields.
\[ \oint_{S} \mathbf{E} \!\cdot\! d\mathbf{A} = \frac{q_{\mathrm{enc}}}{\varepsilon_{0}}, \qquad \nabla \!\cdot\! \mathbf{E} = \frac{\rho}{\varepsilon_{0}}, \qquad \partial_{\mu} J^{\mu}_{\mathrm{coh}} = 0\,. \]
The inverse-square dependence of reflects how curvature coherence distributes evenly across a spherical boundary, preserving constant flux.
Linear superposition ensures multiple charges contribute additively to both and .
Worked Examples
1. Point Charge — Spherical Symmetry
\[ \mathbf{E}(r) = \frac{1}{4\pi\varepsilon_{0}}\,\frac{q}{r^{2}}\,\hat{\mathbf{r}}, \qquad \Phi_{E} = \iint_{S} \mathbf{E}\!\cdot d\mathbf{A} = E(R)\,4\pi R^{2} = \frac{q}{\varepsilon_{0}}\,. \]
For any sphere around a point charge, flux is constant — independent of radius.
2. Infinite Line Charge — Cylindrical Symmetry
\[ \mathbf{E}(r) = \frac{\lambda}{2\pi\varepsilon_{0}\,r}\,\hat{\mathbf{r}}, \qquad \Phi_{E} = E(R)\,(2\pi R L) = \frac{\lambda L}{\varepsilon_{0}}\,. \]
End caps contribute ~0 by symmetry; only the curved side counts.
3. Infinite Sheet — Planar Symmetry
\[ E_{\text{top}} = +\frac{\sigma}{2\varepsilon_{0}}, \qquad E_{\text{bottom}} = -\frac{\sigma}{2\varepsilon_{0}}, \qquad \Phi_{E} = (E_{\text{top}}A + E_{\text{bottom}}A) = \frac{\sigma A}{\varepsilon_{0}}\,. \]
Each face contributes half the total flux; together they yield per side.
Common Pitfalls
1. Surface choice errors
\[ \Phi_{E} = \frac{\sigma A}{\varepsilon_{0}}\,. \]
2. Wrong sign in Poisson or Gauss
\[ \nabla \!\cdot\! \mathbf{E} = \frac{\rho}{\varepsilon_{0}}, \qquad \nabla^{2}\Phi = -\,\frac{\rho}{\varepsilon_{0}}\,. \]
3.No symmetry
\[ \iint_{S} \mathbf{E} \!\cdot\! d\mathbf{A} \;=\; \iiint_{V} (\nabla \!\cdot\! \mathbf{E})\, dV\,. \]
4. Forgetting superposition
\[ \Phi_{\mathrm{total}} = \sum_{i} \Phi_{i}, \qquad \mathbf{E}_{\mathrm{total}} = \sum_{i} \mathbf{E}_{i}, \qquad q_{\mathrm{enc}} = \sum_{i} q_{i}\,. \]
5. Radius scaling confusion
\[ E \propto \frac{1}{R^{2}}, \quad A \propto R^{2} \;\Rightarrow\; \Phi_{E} \text{ is constant.} \]
6. Delta-function sources
\[ \rho(\mathbf{r}) = q\,\delta^{3}(\mathbf{r}) \;\Rightarrow\; q_{\mathrm{enc}} = q\,. \]
7. Units drift
\[ [\mathbf{E}] = \mathrm{V\,m^{-1}}, \quad [\Phi] = \mathrm{V}, \quad [\rho] = \mathrm{C\,m^{-3}}, \quad [\varepsilon_{0}] = \mathrm{F\,m^{-1}}, \quad [\rho_{E}] = \mathrm{J\,m^{-3}}\,. \]
8. Sign of field direction
9. Enclosed vs external charge
Only enclosed \( q_{\mathrm{enc}} \) contributes to flux. External charges may distort \( \mathbf{E} \) but not the total \( \Phi_{E} \).
10. Mixing constants
Always stay in one system: SI or Gaussian, never both.
Coherence View (HST Interpretation)
In HST, Gauss’s law describes the continuity of curvature coherence through a closed boundary:
\[ \partial_{\mu} J^{\mu}_{\mathrm{coh}} = 0 \;\Longleftrightarrow\; \nabla \!\cdot\! \mathbf{E} = \frac{\rho}{\varepsilon_{0}}\,. \]
Charge density marks where coherence curvature deviates from uniform phase.
Field lines are curvature trajectories; equipotential surfaces are constant-tension contours of that coherence field.
Electrostatics is not force through space, but curvature coherence seeking balance.
In essence
Flux is memory, charge is asymmetry, and field is coherence finding its way back to symmetry.
Gauss’s law measures the completeness of that return.
