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The analytical electromagnetics library

Closed-form solutions

What they solve. Why they work.
How to take the equations further.

168 solution records13 families672 study skills7 numerical evaluators
168 solutions
ST-001Exact in a reduced modelhigh confidence

Point, infinite line and uniform sheet charge

Coulomb point-charge fields and Gauss flux law provide the classical starting point. Line and sheet solutions follow from symmetry and are building blocks for capacitance and Green-function methods.

Prescribed charge in a homogeneous unbounded dielectric; a line is infinite along z and a sheet lies at z=0.

Equation actually solved

Laplace/Poisson

Reduction & idealizations

Electrostatic Maxwell system, ∂t=0. Infinite line and sheet have infinite total charge.

Key equations

fields/potentials
ϕq=q4πϵr,Eρ=λ2πϵρ,Ez=σs2ϵsgn⁡z\phi_q=\frac{q}{4\pi\epsilon r},\quad E_\rho=\frac{\lambda}{2\pi\epsilon\rho},\quad E_z=\frac{\sigma_s}{2\epsilon}\operatorname{sgn}z

V; V/m

LaTeX source
\phi_q=\frac{q}{4\pi\epsilon r},\quad E_\rho=\frac{\lambda}{2\pi\epsilon\rho},\quad E_z=\frac{\sigma_s}{2\epsilon}\operatorname{sgn}z

How it is solved

symmetry and Gauss law

Derivation & proof outline

Read with the stated assumptions and cited source. These are explanatory proof steps, not formal proof certificates.

  1. Integrate ∇·D=ρ over a sphere, coaxial cylinder or pillbox.
  2. Symmetry fixes the direction and makes the normal field constant on the integration surface.
  3. A radial gradient has zero curl; the flux jump reproduces each distributional source.

Validity & checks

  • r>0 or ρ>0 away from singular support.
  • Use a finite reference radius for line-charge voltage; zero potential at infinity is unavailable.
Dimensional check

q/(εr) is volts; λ/(ερ) and σs/ε are V/m.

Limiting case

The pillbox flux is σs times area; two opposite sheets cancel outside and add inside.

Expand the equations

Finite sources

  1. Integrate the point-source Green kernel over a bounded segment or patch.
  2. Retain edge fields rather than reusing the infinite-source expression.
  3. The static reduction remains class R even when its representation becomes a series or integral.

Resulting class: R. Read the new assumptions before reuse.

What it is used for

power engineering and lightning

charged conductor and electrode local-field estimates

insight/pedagogy

Adequacy limit: End effects and dielectric interfaces require a different boundary problem.

numerical-method verification and ML surrogates

Point, infinite line and uniform sheet charge reference problem

solver verification benchmark

Adequacy limit: Tests only the stated ideal geometry, source and constitutive model; agreement does not qualify other devices.

Numerical benchmark recipe

q=1 nC; line λ=1 nC/m; evaluate distances 0.1–1 m.

  • Check the stated boundary conditions and equation residual independently.
  • Recover the limiting case: The pillbox flux is σs times area; two opposite sheets cancel outside and add inside.
  • The parameter set is a recommended analytical test case, not a retained native-solver result.

Numerical pitfalls

  • Use SI parameters and exp(jωt); outgoing waves use exp(-jkr).
  • Enforce the validity domain before comparing a numerical solution.

Reusable skill collection

Choose a procedure, then retrieve its versioned YAML contract. A consumer follows the steps and reports the required evidence.

  1. Prescribed charge in a homogeneous unbounded dielectric; a line is infinite along z and a sheet lies at z=0.
  2. Laplace/Poisson
  3. Electrostatic Maxwell system, ∂t=0. Infinite line and sheet have infinite total charge.
  4. symmetry and Gauss law
  5. r>0 or ρ>0 away from singular support.
  6. Use a finite reference radius for line-charge voltage; zero potential at infinity is unavailable.
Download ST-001.describe Download complete entry
Taxonomy & facets
Geometry
Cartesian; cylindrical; spherical
regime
static
source
charge/multipole
medium
isotropic lossless
boundaries
unbounded
method
symmetry and Gauss law
specialFunctions
outputs
fields/potentials

Review notes

  • Analytical reference and proof outline; no native numerical solver execution is claimed.

Sources

  1. Haus and Melcher, Electromagnetic Fields and Energy ↗

    §4.4 Fields Associated with Charge Singularities