Electrostatic Fields, Coulomb Force, and Field Intensity

Unit 2: Electric fields and Gauss’s law

Four lessons: Coulomb’s law and electric superposition, Flux, Gaussian surfaces and symmetry, Conductors, shielding and surface charge, Continuous charge distributions and field integration.

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

Use a nonuniform spherical charge density as a test of the difference between enclosed charge and total charge. For an ideal distribution rho_v(r)=rho0(r/R) inside radius R, integrate 4πr²rho_v(r) dr to obtain Qenclosed(r)=πrho0 r⁴/R. The radial field then scales as r² inside. Compare this result with the linear-in-r field for uniform volume charge. Outside the charged sphere both fields collapse to the same inverse-square expression when the total charges are matched. This is a more convincing use of Gauss’s law than memorizing a few geometries.

Add a conductor-shell scenario. With the field inside metal zero in electrostatic equilibrium, determine what induced surface charges are needed if the cavity contains an isolated point charge. Use conservation of charge on the shell to find charge on its outer surface when the shell began neutral. Finally discuss why an insulating shell does not support the same redistribution and why a finite opening changes perfect shielding. Keep “free charge,” “enclosed charge” and “net total charge” as separate labels throughout.

Guided lesson path

Step 1: Coulomb’s law and electric superposition

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/U02-L01-coulombs-law-and-electric-superposition.html). The mathematical conclusion should follow the boundary and source assumptions, not merely the choice of familiar-looking formula.

Step 2: Flux, Gaussian surfaces and symmetry

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/U02-L02-flux-gaussian-surfaces-and-symmetry.html). The mathematical conclusion should follow the boundary and source assumptions, not merely the choice of familiar-looking formula.

Step 3: Conductors, shielding and surface charge

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/U02-L03-conductors-shielding-and-surface-charge.html). The mathematical conclusion should follow the boundary and source assumptions, not merely the choice of familiar-looking formula.

Step 4: Continuous charge distributions and field integration

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/U02-L04-continuous-charge-distributions-and-field-integration.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-02.html) · [Unit engineering project](../projects/unit-02.html) · [Interactive mathematical model](../simulations/model-02.html).

Unit 2, lesson 1 — Coulomb’s law and electric superposition

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

The electrostatic point-charge law has inverse-square distance dependence because flux spreads over spherical area. Superposition adds the vector contributions of each source; it does not sum magnitudes unless every contribution shares the same direction. For a continuous charge density, an integral replaces the discrete sum. A point charge creates a mathematical singularity at its exact location, so numerical field maps must mask a small neighborhood rather than pretend the plotted value there is finite.

Governing relationships and derivation pathway

For point charges qi at ri, E(r) = (1/(4πepsilon0)) Σ qi (r-ri)/|r-ri|³. A charge qtest experiences F=qtest E. A line charge density lambda has C/m, surface density sigma has C/m², and volume density rho_v has C/m³.

Worked numerical illustration

A +2 nC charge at the origin yields E= kq/r²=179.75 V/m at r=0.10 m, radially outward. Doubling distance lowers E to one quarter; doubling source charge doubles E. This is a useful diagnostic for any particle-field calculator.

Independent numerical variant

For one +1 nC charge at 0.20 m in vacuum, |E|≈224.7 V/m. At 0.40 m its field is 56.2 V/m; the 1/r² scaling is visible.

Engineering context: Two-source cancellation

Place equal positive charges at x=±a and calculate E at the origin: their x components cancel. Then change the left source sign and interpret why the fields add instead. Trace a test-charge path along y=0.1 m and calculate the vector contribution before taking magnitude. In a visualization, never plot the singular field exactly at a charge marker. Practical electrode arrays require iterating source strengths and positions because small charge asymmetries can shift nominally zero-field points.

Sensitivity and design interpretation

Doubling point-source separation at a corresponding scaled observation position reduces field magnitude to one quarter.

Why the model can fail

Electrostatic superposition presumes linear vacuum response. Near real electrodes a prescribed distribution may move because mobile charges redistribute. Distinguish the field calculated for specified sources from a self-consistent conductor equilibrium.

Two resolved conceptual checkpoints

Question A. At what locations can fields from two equal positive charges cancel?

Reasoned answer. At their midpoint the vectors are equal and opposite. Elsewhere equal magnitudes alone do not ensure opposite directions.

Question B. Does doubling all charge strengths double the electrostatic field in the specified-source vacuum model?

Reasoned answer. Yes, the electric field is linear in each qi through superposition; position and geometry are held fixed.

Next steps

[Unit 2 interactive model](../simulations/model-02.html) · [Assessment](../assessments/unit-02.html) · [Project](../projects/unit-02.html) · [Shared study guidance](../STUDY-GUIDE.html).

Unit 2, lesson 2 — Flux, Gaussian surfaces and symmetry

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

Gauss’s law is always true for electric flux through a closed surface, but its use to solve for an unknown field requires symmetry. For a uniformly charged spherical shell, radial field is constant over a concentric sphere. For a very long uniform line, cylindrical side flux survives while flux through end caps vanishes by direction. For a finite line near its end, the infinite-line approximation breaks and a Gaussian cylinder will not have uniform field magnitude on its side.

Governing relationships and derivation pathway

∮S D·dA = Qfree,enclosed; in vacuum D=epsilon0 E. An infinite line gives E=rho_hat lambda/(2πepsilon0 rho); a point charge gives E=r_hat Q/(4πepsilon0 r²). Inside a uniformly charged solid sphere of radius R, E(r)=Q r/(4πepsilon0 R³) for r

Worked numerical illustration

A long wire has lambda=1.0 nC/m. At rho=0.05 m, E=lambda/(2πepsilon0 rho)=359.5 V/m. On a coaxial cylinder of length 0.20 m, the flux of D equals lambda L=0.20 nC, independent of radius.

Independent numerical variant

For lambda=2 nC/m at rho=.1 m, E≈359.5 V/m. A Gaussian cylinder of length .3 m encloses 0.6 nC regardless of the chosen positive radius.

Engineering context: Symmetry audit

Start a Gaussian-flux computation by asking what rotation or translation leaves the entire source distribution unchanged. For a charged sphere, any rotation about the center leaves it unchanged; spherical magnitude can depend only on radius. For a finite rod, translation along its length is not exact symmetry. Construct a flux-balance plot for concentric surfaces and verify that enclosed free charge remains fixed. The curve E(r) may change strongly even when total flux stays constant, because area increases as r squared.

Sensitivity and design interpretation

Doubling distance from an infinite line halves E; doubling radial distance from a point charge reduces E to one quarter.

Why the model can fail

A zero enclosed net charge does not guarantee E=0 everywhere on the surface. Opposite charges may produce strong fields whose signed flux cancels. Gaussian symmetry must be argued, not guessed from the shape of a sketch.

Two resolved conceptual checkpoints

Question A. Can a Gaussian surface surrounding a neutral dipole have nonzero field everywhere?

Reasoned answer. Yes. The signed total D flux is zero, while local normal flux may be positive in some regions and negative in others.

Question B. Why does the long-line field scale as 1/rho rather than 1/rho squared?

Reasoned answer. For a cylinder of fixed length, the area carrying radial flux grows linearly with rho rather than quadratically as spherical area does.

Next steps

[Unit 2 interactive model](../simulations/model-02.html) · [Assessment](../assessments/unit-02.html) · [Project](../projects/unit-02.html) · [Shared study guidance](../STUDY-GUIDE.html).

Unit 2, lesson 3 — Conductors, shielding and surface charge

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

In electrostatic equilibrium, the bulk electric field inside an ideal conductor is zero and its whole connected surface is an equipotential. A nonzero tangential external E would move free charge; therefore the tangential surface component vanishes. Normal displacement has a discontinuity equal to the free surface charge density. A cavity inside a conductor is shielded from static external fields if it has no internal charges and its enclosure is complete; openings and time-dependent fields demand separate analysis.

Governing relationships and derivation pathway

At a conductor–vacuum boundary Dn,out = sigma_free when Ein=0. Therefore En,out = sigma_free/epsilon0. Electrostatic force density can be expressed via the Maxwell stress tensor; for an ideal planar surface the electric pressure is p = epsilon0 E²/2.

Worked numerical illustration

For sigma=1 μC/m² on a smooth ideal conducting surface, immediate exterior En=1e-6/epsilon0≈112,941 V/m. Ideal planar electric pressure is about 0.0565 Pa. These quantities are local idealizations, not direct insulation-safety limits.

Independent numerical variant

For sigma=.5 μC/m², the external vacuum En≈56.5 kV/m on an ideal conductor surface, with direction controlled by the sign of surface charge.

Engineering context: Shield aperture test

Compare an enclosed conductor with a box containing a narrow slot. Static shielding arguments apply cleanly to the complete ideal enclosure but the slot complicates boundary conditions. Draw equipotentials near the opening and discuss why the local field may be enhanced. Use a sensor of finite size to measure a region-averaged field and record distance from the slot. A practical shielding specification should mention frequency range, grounding, seam contact and field orientation instead of a single “perfect shield” claim.

Sensitivity and design interpretation

Doubling free surface charge doubles normal exterior E in the ideal vacuum conductor model.

Why the model can fail

Sharp corners concentrate field; the field of a mathematically perfectly sharp ideal conductor can diverge. A commercial shield must be evaluated for seams, mesh openings, contacts and frequency, not simply labeled a Faraday cage.

Two resolved conceptual checkpoints

Question A. Is the electrostatic field inside an ideal conductor allowed to persist permanently?

Reasoned answer. No. In equilibrium mobile charges redistribute until the interior E vanishes. This does not describe all high-frequency transient or imperfect-conductor cases.

Question B. How does a nonzero free surface charge appear in the boundary condition?

Reasoned answer. The normal component of D jumps by sigma_free. For an ideal conductor in vacuum, the exterior normal field is sigma_free/epsilon0.

Next steps

[Unit 2 interactive model](../simulations/model-02.html) · [Assessment](../assessments/unit-02.html) · [Project](../projects/unit-02.html) · [Shared study guidance](../STUDY-GUIDE.html).

Unit 2, lesson 4 — Continuous charge distributions and field integration

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

Charge on rods, rings, disks and volumes can be treated as many infinitesimal sources. Symmetry often cancels perpendicular components and saves work, but the integration must retain the vector direction until cancellation is justified. On the axis of a uniformly charged ring, transverse components sum to zero while axial components add. Such results are useful checks for approximate numerical quadrature and for estimating electrode fringe fields.

Governing relationships and derivation pathway

For a ring of radius a and total charge Q, axial E_z(z)= (Q/(4πepsilon0)) z/(z²+a²)^(3/2). It vanishes at the center, grows for small positive z and eventually behaves as Q/(4πepsilon0 z²). A finite line is integrated by choosing dq=lambda dl.

Worked numerical illustration

Take Q=5 nC, a=0.10 m, z=0.10 m. E_z = k(5e-9)(0.10)/(0.02)^(3/2)≈1,589 V/m, directed along +z for positive charge. A point-charge replacement at z=a would be a poor near-field approximation.

Independent numerical variant

For Q=2 nC, a=.1 m, z=.2 m, the axial ring field is about 322 V/m from kQz/(a²+z²)^(3/2); check the units of the denominator carefully.

Engineering context: Ring model validation

Numerically approximate the ring with 12, 48 and 192 equal charge segments and compute the on-axis field. Compare to the continuous analytical expression. At z=0 the field cancels; at very large z it approaches the point-charge limit. Distinguish quadrature error from model error: adding segments cannot fix an approximation that treats a thick nonuniform torus as a perfectly thin uniform ring. Plot the relative error as a function of both segment count and observation distance.

Sensitivity and design interpretation

On a ring axis, far-field E falls as z^-2, but close to the center the axial E grows approximately proportionally to z.

Why the model can fail

Do not replace distributed charge by total charge at its center unless distance is large compared with its extent and approximation error is acceptable. Check limiting cases z→0 and z≫a before trusting any symbolic integration.

Two resolved conceptual checkpoints

Question A. What is the far-axis limit of the ring field?

Reasoned answer. For z much larger than a, E_z approaches kQ/z² because z/(z²+a²)^(3/2) approaches 1/z².

Question B. Why is the ring-axis field zero at its center even when charge is nonzero?

Reasoned answer. Each charge element has an opposing transverse contribution. The axial factor z is zero at the center.

Next steps

[Unit 2 interactive model](../simulations/model-02.html) · [Assessment](../assessments/unit-02.html) · [Project](../projects/unit-02.html) · [Shared study guidance](../STUDY-GUIDE.html).

Unit 2 engineering project: Gaussian symmetry and shielding decision

Problem statement and intended decision

A sensor body is approximated by a spherical shell with a concentric charge distribution. Calculate radial E in the empty interior and outside, then change the enclosed charge by a known increment. Contrast a full closed metal enclosure with one containing a measurement opening. Provide an explicit list of idealizations: electrostatics, smooth surface, perfect conductor, isolated source and no unmodeled grounding path. The project decision is whether Gauss’s law alone determines a local field after symmetry is broken. Explain why the law remains valid even when it no longer solves for E directly.

Detailed investigation plan

Use a nonuniform spherical charge density as a test of the difference between enclosed charge and total charge. For an ideal distribution rho_v(r)=rho0(r/R) inside radius R, integrate 4πr²rho_v(r) dr to obtain Qenclosed(r)=πrho0 r⁴/R. The radial field then scales as r² inside. Compare this result with the linear-in-r field for uniform volume charge. Outside the charged sphere both fields collapse to the same inverse-square expression when the total charges are matched. This is a more convincing use of Gauss’s law than memorizing a few geometries.

Add a conductor-shell scenario. With the field inside metal zero in electrostatic equilibrium, determine what induced surface charges are needed if the cavity contains an isolated point charge. Use conservation of charge on the shell to find charge on its outer surface when the shell began neutral. Finally discuss why an insulating shell does not support the same redistribution and why a finite opening changes perfect shielding. Keep “free charge,” “enclosed charge” and “net total charge” as separate labels throughout.

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 2 assessment — Electric fields and Gauss’s law

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 — Coulomb’s law and electric superposition

Prompt: At what locations can fields from two equal positive charges cancel?

Reveal worked reasoning

At their midpoint the vectors are equal and opposite. Elsewhere equal magnitudes alone do not ensure opposite directions. The matching lesson and shared study guide provide additional verification questions.

Question 2 — Coulomb’s law and electric superposition

Prompt: Does doubling all charge strengths double the electrostatic field in the specified-source vacuum model?

Reveal worked reasoning

Yes, the electric field is linear in each qi through superposition; position and geometry are held fixed. The matching lesson and shared study guide provide additional verification questions.

Question 3 — Coulomb’s law and electric superposition numerical and decision challenge

Prompt: Independently derive and check this scenario: For one +1 nC charge at 0.20 m in vacuum, |E|≈224.7 V/m. At 0.40 m its field is 56.2 V/m; the 1/r² scaling is visible. Then choose a parameter to change, predict its effect from doubling point-source separation at a corresponding scaled observation position reduces field magnitude to one quarter. State which quantities remain fixed and identify one important excluded physical effect.

Reveal worked reasoning and review criteria

Reference calculation: For one +1 nC charge at 0.20 m in vacuum, |E|≈224.7 V/m. At 0.40 m its field is 56.2 V/m; the 1/r² scaling is visible.

Sensitivity logic: Doubling point-source separation at a corresponding scaled observation position reduces field magnitude to one quarter.

Engineering qualification: Electrostatic superposition presumes linear vacuum response. Near real electrodes a prescribed distribution may move because mobile charges redistribute. Distinguish the field calculated for specified sources from a self-consistent conductor equilibrium. 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 — Flux, Gaussian surfaces and symmetry

Prompt: Can a Gaussian surface surrounding a neutral dipole have nonzero field everywhere?

Reveal worked reasoning

Yes. The signed total D flux is zero, while local normal flux may be positive in some regions and negative in others. The matching lesson and shared study guide provide additional verification questions.

Question 5 — Flux, Gaussian surfaces and symmetry

Prompt: Why does the long-line field scale as 1/rho rather than 1/rho squared?

Reveal worked reasoning

For a cylinder of fixed length, the area carrying radial flux grows linearly with rho rather than quadratically as spherical area does. The matching lesson and shared study guide provide additional verification questions.

Question 6 — Flux, Gaussian surfaces and symmetry numerical and decision challenge

Prompt: Independently derive and check this scenario: For lambda=2 nC/m at rho=.1 m, E≈359.5 V/m. A Gaussian cylinder of length .3 m encloses 0.6 nC regardless of the chosen positive radius. Then choose a parameter to change, predict its effect from doubling distance from an infinite line halves E; doubling radial distance from a point charge reduces E to one quarter. State which quantities remain fixed and identify one important excluded physical effect.

Reveal worked reasoning and review criteria

Reference calculation: For lambda=2 nC/m at rho=.1 m, E≈359.5 V/m. A Gaussian cylinder of length .3 m encloses 0.6 nC regardless of the chosen positive radius.

Sensitivity logic: Doubling distance from an infinite line halves E; doubling radial distance from a point charge reduces E to one quarter.

Engineering qualification: A zero enclosed net charge does not guarantee E=0 everywhere on the surface. Opposite charges may produce strong fields whose signed flux cancels. Gaussian symmetry must be argued, not guessed from the shape of a sketch. 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 — Conductors, shielding and surface charge

Prompt: Is the electrostatic field inside an ideal conductor allowed to persist permanently?

Reveal worked reasoning

No. In equilibrium mobile charges redistribute until the interior E vanishes. This does not describe all high-frequency transient or imperfect-conductor cases. The matching lesson and shared study guide provide additional verification questions.

Question 8 — Conductors, shielding and surface charge

Prompt: How does a nonzero free surface charge appear in the boundary condition?

Reveal worked reasoning

The normal component of D jumps by sigma_free. For an ideal conductor in vacuum, the exterior normal field is sigma_free/epsilon0. The matching lesson and shared study guide provide additional verification questions.

Question 9 — Conductors, shielding and surface charge numerical and decision challenge

Prompt: Independently derive and check this scenario: For sigma=.5 μC/m², the external vacuum En≈56.5 kV/m on an ideal conductor surface, with direction controlled by the sign of surface charge. Then choose a parameter to change, predict its effect from doubling free surface charge doubles normal exterior E in the ideal vacuum conductor model. State which quantities remain fixed and identify one important excluded physical effect.

Reveal worked reasoning and review criteria

Reference calculation: For sigma=.5 μC/m², the external vacuum En≈56.5 kV/m on an ideal conductor surface, with direction controlled by the sign of surface charge.

Sensitivity logic: Doubling free surface charge doubles normal exterior E in the ideal vacuum conductor model.

Engineering qualification: Sharp corners concentrate field; the field of a mathematically perfectly sharp ideal conductor can diverge. A commercial shield must be evaluated for seams, mesh openings, contacts and frequency, not simply labeled a Faraday cage. 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 — Continuous charge distributions and field integration

Prompt: What is the far-axis limit of the ring field?

Reveal worked reasoning

For z much larger than a, E_z approaches kQ/z² because z/(z²+a²)^(3/2) approaches 1/z². The matching lesson and shared study guide provide additional verification questions.

Question 11 — Continuous charge distributions and field integration

Prompt: Why is the ring-axis field zero at its center even when charge is nonzero?

Reveal worked reasoning

Each charge element has an opposing transverse contribution. The axial factor z is zero at the center. The matching lesson and shared study guide provide additional verification questions.

Question 12 — Continuous charge distributions and field integration numerical and decision challenge

Prompt: Independently derive and check this scenario: For Q=2 nC, a=.1 m, z=.2 m, the axial ring field is about 322 V/m from kQz/(a²+z²)^(3/2); check the units of the denominator carefully. Then choose a parameter to change, predict its effect from on a ring axis, far-field E falls as z^-2, but close to the center the axial E grows approximately proportionally to z. State which quantities remain fixed and identify one important excluded physical effect.

Reveal worked reasoning and review criteria

Reference calculation: For Q=2 nC, a=.1 m, z=.2 m, the axial ring field is about 322 V/m from kQz/(a²+z²)^(3/2); check the units of the denominator carefully.

Sensitivity logic: On a ring axis, far-field E falls as z^-2, but close to the center the axial E grows approximately proportionally to z.

Engineering qualification: Do not replace distributed charge by total charge at its center unless distance is large compared with its extent and approximation error is acceptable. Check limiting cases z→0 and z≫a before trusting any symbolic integration. 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.