Gauss Law, Electric Flux Density, and Divergence

Unit 3: Electric potential and boundary-value problems

Four lessons: Potential, work and path independence, Poisson’s and Laplace’s equations, Uniqueness and boundary conditions, Finite-difference electrostatics and mesh checks.

Learning outcomes

By the end of this unit, explain the physical field variables and their SI units; solve a useful analytical reference geometry; reproduce a field-level numerical calculation; identify boundary assumptions; and explain an engineering decision that would be unjustified by the ideal model alone.

Unit investigation

Implement a 2D five-point-stencil potential solver, starting with a region with fixed left and right electrode potentials and insulated top and bottom boundaries. Because top and bottom are Neumann boundaries, carefully mirror or difference the normal derivative so it vanishes; do not silently clamp them to zero volts. Compare with the 1D analytical linear ramp. Define a residual R_ij=4V_ij-(V_{i+1,j}+V_{i-1,j}+V_{i,j+1}+V_{i,j-1}). Show both the maximum residual and max deviation from the benchmark. Refining the grid should not change the exact ideal ramp beyond roundoff and incomplete iteration.

Introduce a dielectric interface with a different εr on one half. The stencil must change to preserve normal D continuity; using simple unweighted arithmetic averages across the interface is incorrect. Derive the face-flux balance using conductance-like coefficients epsilon_face/h². Compare grid solutions with a hand-calculated one-dimensional layered slab. Plot normal E on both sides of the interface; its discontinuity is physical while Dn remains continuous without free interfacial charge. Report grid dimensions, dielectric assumptions, stopping tolerance and an energy estimate to enable reproducibility.

Guided lesson path

Step 1: Potential, work and path independence

Review the governing relationship, reproduce the numerical variant with independent unit conversions, and explicitly note one observation that would falsify the model's assumptions. See [lesson 1](../lessons/U03-L01-potential-work-and-path-independence.html). The mathematical conclusion should follow the boundary and source assumptions, not merely the choice of familiar-looking formula.

Step 2: Poisson’s and Laplace’s equations

Review the governing relationship, reproduce the numerical variant with independent unit conversions, and explicitly note one observation that would falsify the model's assumptions. See [lesson 2](../lessons/U03-L02-poissons-and-laplaces-equations.html). The mathematical conclusion should follow the boundary and source assumptions, not merely the choice of familiar-looking formula.

Step 3: Uniqueness and boundary conditions

Review the governing relationship, reproduce the numerical variant with independent unit conversions, and explicitly note one observation that would falsify the model's assumptions. See [lesson 3](../lessons/U03-L03-uniqueness-and-boundary-conditions.html). The mathematical conclusion should follow the boundary and source assumptions, not merely the choice of familiar-looking formula.

Step 4: Finite-difference electrostatics and mesh checks

Review the governing relationship, reproduce the numerical variant with independent unit conversions, and explicitly note one observation that would falsify the model's assumptions. See [lesson 4](../lessons/U03-L04-finite-difference-electrostatics-and-mesh-checks.html). The mathematical conclusion should follow the boundary and source assumptions, not merely the choice of familiar-looking formula.

Unit verification checklist

• Define at least one coordinate system and positive contour or normal orientation.
• State which source or boundary quantities are controlled and which are outputs.
• Compare analytical and computational predictions, including one limiting case.
• Report at least one sensitivity result rather than a single nominal value.
• Describe which material data would be needed for genuine engineering validation.
• Prepare a concise error budget identifying unit conversion, approximate geometry and material uncertainty.

Practice and project: [Unit assessment](../assessments/unit-03.html) · [Unit engineering project](../projects/unit-03.html) · [Interactive mathematical model](../simulations/model-03.html).

Unit 3, lesson 1 — Potential, work and path independence

Book 21 — Electromagnetic Field Theory I · Prerequisites: multivariable calculus, introductory circuits and SI units. Quality note: Original educational draft; physical equations and instructional solutions require independent review before publication.

Physical idea and meaning

Electric potential translates a conservative electric field into energy per unit charge. The voltage difference between two points equals minus the line integral of E along a path. In electrostatics the result is path-independent, so a scalar potential is defined up to an arbitrary constant. Engineers set a useful reference such as V=0 at infinity for localized charge or at a selected ground electrode. Ground is a boundary condition, not a physical sink for an unlimited amount of charge.

Governing relationships and derivation pathway

V(B)-V(A) = -∫A→B E·dl. For a point source V(r)=Q/(4πepsilon0 r) when infinity is the reference, and E=-grad V. Work done by the electric field on charge q is q[V(A)-V(B)].

Worked numerical illustration

Between ideal plates 10 mm apart with V(top)=100 V and V(bottom)=0 V, ignoring fringing, |E|=100/0.010=10,000 V/m. The field points from the high-potential plate toward the low-potential plate.

Independent numerical variant

Across ideal plates 40 V apart by 4 mm, E magnitude is 10 kV/m. Moving +1 nC down the potential by 40 V releases field work 40 nJ.

Engineering context: Measuring work from voltages

A voltmeter reports a difference, not a locally unique absolute potential. Choose a reference node and measure two points along a nominally uniform parallel-plate region. The estimated electric field is the negative change of voltage divided by distance. Repeat the test with reversed probes; the display sign reverses but the physical field does not. Near the plate edge, equipotential contours bend and the simple voltage-spacing model should be explicitly labeled an approximation.

Sensitivity and design interpretation

Doubling parallel-plate separation at fixed voltage halves E, whereas doubling voltage at fixed separation doubles E.

Why the model can fail

A negative test charge moves toward higher potential if it follows the electrostatic force. Potential increases and decreases should not be confused with potential energy, which is U=qV and changes sign with q.

Two resolved conceptual checkpoints

Question A. How does E relate to the spacing of equipotential contours?

Reasoned answer. Where nearby equipotentials differ by the same voltage across a shorter distance,

Question B. Does choosing a different ground reference alter force between charges?

Reasoned answer. No. Adding a constant to V leaves grad V unchanged, so E and electrostatic force are unaffected.

Next steps

[Unit 3 interactive model](../simulations/model-03.html) · [Assessment](../assessments/unit-03.html) · [Project](../projects/unit-03.html) · [Shared study guidance](../STUDY-GUIDE.html).

Unit 3, lesson 2 — Poisson’s and Laplace’s equations

Book 21 — Electromagnetic Field Theory I · Prerequisites: multivariable calculus, introductory circuits and SI units. Quality note: Original educational draft; physical equations and instructional solutions require independent review before publication.

Physical idea and meaning

Combining Gauss’s law with E=-grad V gives an equation for potential. Poisson’s equation includes local charge density; Laplace’s equation applies where charge density vanishes in a homogeneous medium. These are not merely algebra exercises: electrode geometry is encoded through boundary conditions, and the resulting potential predicts field concentrations that govern dielectric failure, sensor sensitivity and capacitance.

Governing relationships and derivation pathway

In a uniform isotropic dielectric, div(epsilon grad V)=-rho_free; if epsilon is constant, ∇²V=-rho_free/epsilon. In charge-free homogeneous material ∇²V=0. Cartesian Laplacian is ∂²V/∂x²+∂²V/∂y²+∂²V/∂z².

Worked numerical illustration

For one-dimensional plate geometry with no space charge, d²V/dx²=0. Imposing V(0)=0 and V(d)=V0 gives V(x)=V0 x/d and uniform E_x=-V0/d. This simple derivation is the reference against which finite-difference plate solutions should be tested.

Independent numerical variant

In a charge-free 1D region with V(0)=10 V and V(.02)=30 V, V(x)=10+1000x volts and E=-1000 V/m throughout.

Engineering context: Numerical Laplacian benchmark

A finite-difference solver can reproduce a straight voltage ramp between ideal infinite electrodes without spatial charge. This is the first correctness benchmark; only after passing it should the solver handle more complicated shapes. Compare mean-square residual with maximum field error and report both. A model with potential nearly correct at every node may still give inaccurate E near a singular boundary, because differentiation amplifies grid-level errors. That distinction is essential when predicting dielectric stress.

Sensitivity and design interpretation

Doubling all lengths while preserving electrode voltages approximately halves E for geometrically similar charge-free Laplace solutions.

Why the model can fail

Laplace’s equation does not say potential is constant; linear and curved harmonic solutions exist. At an interface between unequal permittivities, one must apply boundary conditions rather than continue the same single-material equation blindly.

Two resolved conceptual checkpoints

Question A. Why does no volume free charge produce Laplace rather than Poisson?

Reasoned answer. With rho_free=0 in a homogeneous linear dielectric, the source term of Poisson vanishes. Boundary voltages still generate nonzero V and E.

Question B. What mistake results from applying an equal-spacing five-point stencil across sharply different ε?

Reasoned answer. It ignores flux continuity; a conservative face-weighted discretization is required at material interfaces.

Next steps

[Unit 3 interactive model](../simulations/model-03.html) · [Assessment](../assessments/unit-03.html) · [Project](../projects/unit-03.html) · [Shared study guidance](../STUDY-GUIDE.html).

Unit 3, lesson 3 — Uniqueness and boundary conditions

Book 21 — Electromagnetic Field Theory I · Prerequisites: multivariable calculus, introductory circuits and SI units. Quality note: Original educational draft; physical equations and instructional solutions require independent review before publication.

Physical idea and meaning

A field solution is useful only if its sources and boundaries are specified. Dirichlet conditions prescribe potential; Neumann conditions prescribe normal potential derivative or displacement flux. Mixed conditions are common on real structures. For a well-posed electrostatic problem, the uniqueness theorem justifies comparison between a guessed analytical solution and a numerical solver: if they satisfy the governing equation and the complete boundary data, they represent the same physical solution.

Governing relationships and derivation pathway

Dirichlet: V|boundary = known function. Neumann: -epsilon ∂V/∂n = specified Dn. Interface conditions: tangential E continuous in electrostatics; jump in Dn equals free surface charge. Pure Neumann problems generally determine V only up to an additive constant and require compatibility of total flux with enclosed free charge.

Worked numerical illustration

Consider a metal enclosure set to 0 V with a central electrode at 10 V. Changing only the outer wall to 5 V shifts field differences if the inner remains at 10 V; raising both by 5 V leaves E unchanged. This isolates absolute-potential references from actual voltage differences.

Independent numerical variant

If both electrodes shift from (0,20) V to (100,120) V, E remains unchanged. If only the high electrode shifts to 120 V, E changes fivefold relative to the original 20 V difference.

Engineering context: Floating conductors

Imagine a metal insert inside an insulating enclosure. If the insert is insulated from external ground, its potential cannot simply be fixed to zero. Instead, solve an unknown constant potential that also satisfies its net-charge condition. The distinction affects capacitance measurements and capacitive sensor calibration. When reporting results, give conductor identities, prescribed potentials, floating-charge constraints and permittivities. Two images of the same geometry can represent different boundary-value problems if those labels differ.

Sensitivity and design interpretation

Raising both prescribed electrode voltages by the same constant leaves E unchanged, but changing only one boundary changes the field.

Why the model can fail

A floating conductor cannot be modeled automatically as a prescribed zero-voltage conductor; its net charge and unknown potential must be solved together. Numerical instability near corners may come from boundary idealization, not a programming bug.

Two resolved conceptual checkpoints

Question A. What extra constraint is required for a floating conductor?

Reasoned answer. Its unknown constant potential must be solved together with the total charge condition. Setting it to ground changes the physical problem.

Question B. What information does a pure-Neumann formulation fail to determine?

Reasoned answer. An arbitrary additive constant of potential remains; charge compatibility and a reference convention are needed.

Next steps

[Unit 3 interactive model](../simulations/model-03.html) · [Assessment](../assessments/unit-03.html) · [Project](../projects/unit-03.html) · [Shared study guidance](../STUDY-GUIDE.html).

Unit 3, lesson 4 — Finite-difference electrostatics and mesh checks

Book 21 — Electromagnetic Field Theory I · Prerequisites: multivariable calculus, introductory circuits and SI units. Quality note: Original educational draft; physical equations and instructional solutions require independent review before publication.

Physical idea and meaning

For complex cross sections, replace derivatives by finite differences on a grid. At an interior uniform-grid node in a two-dimensional charge-free region, the potential equals the average of its four nearest neighbors. Successive over-relaxation can accelerate convergence, but a small residual is not proof that the geometry is resolved. Mesh refinement and a known analytical benchmark are needed to separate iteration error from discretization error.

Governing relationships and derivation pathway

With grid spacing h and homogeneous epsilon, V(i,j)≈[V(i+1,j)+V(i-1,j)+V(i,j+1)+V(i,j-1)]/4. For nonuniform epsilon use a conservative face-flux discretization of div(epsilon grad V), not a naïve constant-epsilon stencil.

Worked numerical illustration

On a 1D plate with 100 V across 5 equal steps, exact interior potentials are 20,40,60,80 V. A solver returning 19.9,40.1,60.0,79.8 V may have converged poorly or used inconsistent boundaries; report max absolute deviation.

Independent numerical variant

A 1D plate split into ten equal 1 mm cells with 100 V across it has node potentials 0,10,...,100 V. The residual of the constant-step second difference is zero in exact arithmetic.

Engineering context: Residual versus convergence

At each relaxation iteration, evaluate the local discrete Laplacian residual and track its maximum. Also compare successive meshes with h, h/2 and h/4 to estimate resolution error. Use the plate benchmark, an analytical concentric-cylinder case and a charge-conservation check. If the result changes by several percent as the mesh is refined, it should not be advertised as an accurate field prediction. Grid size and termination tolerance belong in a reproducible numerical result.

Sensitivity and design interpretation

Halving mesh spacing increases the number of interior nodes by approximately four in a two-dimensional regular domain.

Why the model can fail

In a conductor approximation, imposing both potential and normal derivative on the same boundary generally overconstrains the model. Check global conservation: total outward displacement flux should match enclosed free charge.

Two resolved conceptual checkpoints

Question A. If a numerical residual is tiny but peak E changes on mesh refinement, is the model validated?

Reasoned answer. No. Iteration convergence is not spatial convergence; the grid may not resolve the local gradient or corner field.

Question B. Why does differentiating V amplify discretization noise?

Reasoned answer. The finite-difference derivative divides voltage differences by grid spacing, so small potential errors can create disproportionately large field errors.

Next steps

[Unit 3 interactive model](../simulations/model-03.html) · [Assessment](../assessments/unit-03.html) · [Project](../projects/unit-03.html) · [Shared study guidance](../STUDY-GUIDE.html).

Unit 3 engineering project: Potential solver for a capacitor cross-section

Problem statement and intended decision

Construct a charge-free 2D rectangular domain containing two electrode boundaries at 0 and 20 V. Start with a 1D parallel-plate benchmark and then add a central notch to one electrode. Use a five-point finite-difference stencil away from dielectric interfaces and enforce prescribed boundary nodes every iteration. Report max potential residual, peak field location, and a three-level mesh convergence comparison. The engineering goal is to locate candidate high-stress corners, not to claim a certified breakdown voltage from an unresolved geometric singularity.

Detailed investigation plan

Implement a 2D five-point-stencil potential solver, starting with a region with fixed left and right electrode potentials and insulated top and bottom boundaries. Because top and bottom are Neumann boundaries, carefully mirror or difference the normal derivative so it vanishes; do not silently clamp them to zero volts. Compare with the 1D analytical linear ramp. Define a residual R_ij=4V_ij-(V_{i+1,j}+V_{i-1,j}+V_{i,j+1}+V_{i,j-1}). Show both the maximum residual and max deviation from the benchmark. Refining the grid should not change the exact ideal ramp beyond roundoff and incomplete iteration.

Introduce a dielectric interface with a different εr on one half. The stencil must change to preserve normal D continuity; using simple unweighted arithmetic averages across the interface is incorrect. Derive the face-flux balance using conductance-like coefficients epsilon_face/h². Compare grid solutions with a hand-calculated one-dimensional layered slab. Plot normal E on both sides of the interface; its discontinuity is physical while Dn remains continuous without free interfacial charge. Report grid dimensions, dielectric assumptions, stopping tolerance and an energy estimate to enable reproducibility.

Submission requirements

Follow the seven evidence deliverables and the 100-point project rubric in the [Shared study guidance](../STUDY-GUIDE.html). The unit-specific source model, investigation plan, parameter choices and physical limitations above determine whether the report is adequate.

Submission safeguard

All activities are simulation and analysis only. If a project might ordinarily be performed with mains current, charged capacitors, electromagnets, high-voltage insulation or RF power, replace hardware with the ideal mathematical model and a synthetic dataset. Do not treat this project brief as authorization or guidance to perform dangerous laboratory measurements.

Unit 3 assessment — Electric potential and boundary-value problems

Instructions: Work independently before expanding answers. These twelve questions cover all four lessons. State your coordinate reference, units, sign convention and physical assumptions. The answers below explain the expected reasoning; identical numerical output alone is insufficient.

Question 1 — Potential, work and path independence

Prompt: How does E relate to the spacing of equipotential contours?

Reveal worked reasoning

Where nearby equipotentials differ by the same voltage across a shorter distance, |E| is larger; E points perpendicular toward decreasing V. The matching lesson and shared study guide provide additional verification questions.

Question 2 — Potential, work and path independence

Prompt: Does choosing a different ground reference alter force between charges?

Reveal worked reasoning

No. Adding a constant to V leaves grad V unchanged, so E and electrostatic force are unaffected. The matching lesson and shared study guide provide additional verification questions.

Question 3 — Potential, work and path independence numerical and decision challenge

Prompt: Independently derive and check this scenario: Across ideal plates 40 V apart by 4 mm, E magnitude is 10 kV/m. Moving +1 nC down the potential by 40 V releases field work 40 nJ. Then choose a parameter to change, predict its effect from doubling parallel-plate separation at fixed voltage halves E, whereas doubling voltage at fixed separation doubles E. State which quantities remain fixed and identify one important excluded physical effect.

Reveal worked reasoning and review criteria

Reference calculation: Across ideal plates 40 V apart by 4 mm, E magnitude is 10 kV/m. Moving +1 nC down the potential by 40 V releases field work 40 nJ.

Sensitivity logic: Doubling parallel-plate separation at fixed voltage halves E, whereas doubling voltage at fixed separation doubles E.

Engineering qualification: A negative test charge moves toward higher potential if it follows the electrostatic force. Potential increases and decreases should not be confused with potential energy, which is U=qV and changes sign with q. The final report must include a quantity with correct SI units, an analytical limit and a specific reason the ideal calculation is not sufficient for hardware sign-off.

Question 4 — Poisson’s and Laplace’s equations

Prompt: Why does no volume free charge produce Laplace rather than Poisson?

Reveal worked reasoning

With rho_free=0 in a homogeneous linear dielectric, the source term of Poisson vanishes. Boundary voltages still generate nonzero V and E. The matching lesson and shared study guide provide additional verification questions.

Question 5 — Poisson’s and Laplace’s equations

Prompt: What mistake results from applying an equal-spacing five-point stencil across sharply different ε?

Reveal worked reasoning

It ignores flux continuity; a conservative face-weighted discretization is required at material interfaces. The matching lesson and shared study guide provide additional verification questions.

Question 6 — Poisson’s and Laplace’s equations numerical and decision challenge

Prompt: Independently derive and check this scenario: In a charge-free 1D region with V(0)=10 V and V(.02)=30 V, V(x)=10+1000x volts and E=-1000 V/m throughout. Then choose a parameter to change, predict its effect from doubling all lengths while preserving electrode voltages approximately halves E for geometrically similar charge-free Laplace solutions. State which quantities remain fixed and identify one important excluded physical effect.

Reveal worked reasoning and review criteria

Reference calculation: In a charge-free 1D region with V(0)=10 V and V(.02)=30 V, V(x)=10+1000x volts and E=-1000 V/m throughout.

Sensitivity logic: Doubling all lengths while preserving electrode voltages approximately halves E for geometrically similar charge-free Laplace solutions.

Engineering qualification: Laplace’s equation does not say potential is constant; linear and curved harmonic solutions exist. At an interface between unequal permittivities, one must apply boundary conditions rather than continue the same single-material equation blindly. The final report must include a quantity with correct SI units, an analytical limit and a specific reason the ideal calculation is not sufficient for hardware sign-off.

Question 7 — Uniqueness and boundary conditions

Prompt: What extra constraint is required for a floating conductor?

Reveal worked reasoning

Its unknown constant potential must be solved together with the total charge condition. Setting it to ground changes the physical problem. The matching lesson and shared study guide provide additional verification questions.

Question 8 — Uniqueness and boundary conditions

Prompt: What information does a pure-Neumann formulation fail to determine?

Reveal worked reasoning

An arbitrary additive constant of potential remains; charge compatibility and a reference convention are needed. The matching lesson and shared study guide provide additional verification questions.

Question 9 — Uniqueness and boundary conditions numerical and decision challenge

Prompt: Independently derive and check this scenario: If both electrodes shift from (0,20) V to (100,120) V, E remains unchanged. If only the high electrode shifts to 120 V, E changes fivefold relative to the original 20 V difference. Then choose a parameter to change, predict its effect from raising both prescribed electrode voltages by the same constant leaves E unchanged, but changing only one boundary changes the field. State which quantities remain fixed and identify one important excluded physical effect.

Reveal worked reasoning and review criteria

Reference calculation: If both electrodes shift from (0,20) V to (100,120) V, E remains unchanged. If only the high electrode shifts to 120 V, E changes fivefold relative to the original 20 V difference.

Sensitivity logic: Raising both prescribed electrode voltages by the same constant leaves E unchanged, but changing only one boundary changes the field.

Engineering qualification: A floating conductor cannot be modeled automatically as a prescribed zero-voltage conductor; its net charge and unknown potential must be solved together. Numerical instability near corners may come from boundary idealization, not a programming bug. The final report must include a quantity with correct SI units, an analytical limit and a specific reason the ideal calculation is not sufficient for hardware sign-off.

Question 10 — Finite-difference electrostatics and mesh checks

Prompt: If a numerical residual is tiny but peak E changes on mesh refinement, is the model validated?

Reveal worked reasoning

No. Iteration convergence is not spatial convergence; the grid may not resolve the local gradient or corner field. The matching lesson and shared study guide provide additional verification questions.

Question 11 — Finite-difference electrostatics and mesh checks

Prompt: Why does differentiating V amplify discretization noise?

Reveal worked reasoning

The finite-difference derivative divides voltage differences by grid spacing, so small potential errors can create disproportionately large field errors. The matching lesson and shared study guide provide additional verification questions.

Question 12 — Finite-difference electrostatics and mesh checks numerical and decision challenge

Prompt: Independently derive and check this scenario: A 1D plate split into ten equal 1 mm cells with 100 V across it has node potentials 0,10,...,100 V. The residual of the constant-step second difference is zero in exact arithmetic. Then choose a parameter to change, predict its effect from halving mesh spacing increases the number of interior nodes by approximately four in a two-dimensional regular domain. State which quantities remain fixed and identify one important excluded physical effect.

Reveal worked reasoning and review criteria

Reference calculation: A 1D plate split into ten equal 1 mm cells with 100 V across it has node potentials 0,10,...,100 V. The residual of the constant-step second difference is zero in exact arithmetic.

Sensitivity logic: Halving mesh spacing increases the number of interior nodes by approximately four in a two-dimensional regular domain.

Engineering qualification: In a conductor approximation, imposing both potential and normal derivative on the same boundary generally overconstrains the model. Check global conservation: total outward displacement flux should match enclosed free charge. The final report must include a quantity with correct SI units, an analytical limit and a specific reason the ideal calculation is not sufficient for hardware sign-off.