Poisson and Laplace Equations and Boundary-Value Methods

Unit 6: Magnetostatic fields and magnetic materials

Four lessons: Magnetic flux density and the Biot–Savart law, Ampère’s law and symmetry, Magnetic materials and field boundaries, Vector potential and magnetic flux conservation.

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

Compute the field at a set of points near two parallel opposite currents. Use Biot–Savart superposition in the infinite-wire approximation to draw B vectors, but mark the assumptions that the wires are straight, very long and sufficiently separated from terminations. Use Ampère’s integral law along a loop enclosing only one conductor and then a loop enclosing both. The second has zero net enclosed current, yet local B on its perimeter is not generally zero. This is an accessible example of the danger in interpreting an integral law as a pointwise zero-field prediction.

Turn the analysis into an EMC routing choice. A paired current and close return typically gives a smaller loop area than two widely separated conductors; compare field magnitude at specified observation points. Avoid interpreting the field as an absolute safety rating; near-field exposure depends on time waveform, geometry and applicable measurement standards. Include vector arrows and one independently checked point from direct analytic formulas.

Guided lesson path

Step 1: Magnetic flux density and the Biot–Savart law

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/U06-L01-magnetic-flux-density-and-the-biot-savart-law.html). The mathematical conclusion should follow the boundary and source assumptions, not merely the choice of familiar-looking formula.

Step 2: Ampère’s law 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/U06-L02-amp-res-law-and-symmetry.html). The mathematical conclusion should follow the boundary and source assumptions, not merely the choice of familiar-looking formula.

Step 3: Magnetic materials and field boundaries

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/U06-L03-magnetic-materials-and-field-boundaries.html). The mathematical conclusion should follow the boundary and source assumptions, not merely the choice of familiar-looking formula.

Step 4: Vector potential and magnetic flux conservation

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/U06-L04-vector-potential-and-magnetic-flux-conservation.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-06.html) · [Unit engineering project](../projects/unit-06.html) · [Interactive mathematical model](../simulations/model-06.html).

Unit 6, lesson 1 — Magnetic flux density and the Biot–Savart law

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 stationary electric charge produces an electrostatic field, while a steady current produces a magnetostatic field. Biot–Savart calculates B by adding contributions from current elements, with orientation determined by a cross product. The result is not generally radial: around a long straight wire, B wraps circumferentially. Where a conductor has finite thickness, the singular filament formula is not valid inside the metal; use the appropriate current distribution.

Governing relationships and derivation pathway

For a current element I dl, dB=μ0/(4π) I (dl×Rhat)/R². For an infinitely long thin straight wire, B_phi=μ0 I/(2πrho). For a circular loop of radius a, B_z(0)=μ0 I/(2a).

Worked numerical illustration

A long straight wire carrying 5 A produces B≈(4π×10^-7 ×5)/(2π×0.02)=50 μT at 2 cm. The right-hand rule determines phi direction; reversing current reverses B.

Independent numerical variant

A long 8 A wire at 4 cm generates B=40 μT in vacuum, assuming the observation lies outside its thin cross section.

Engineering context: Wire-return geometry

Plot the B field around one long straight current and then introduce a parallel return conductor carrying equal opposite current. Far away, the return path cancels much of the pair’s magnetic field; near one wire, the local field remains significant. This illustrates why a measured magnetic field cannot be predicted from a supply wire alone. In compact electronics routing, loop area rather than wire current alone often controls magnetic coupling and emission.

Sensitivity and design interpretation

Doubling straight-wire current doubles B; doubling observation radius halves B outside a thin long wire.

Why the model can fail

A finite conductor needs an end-angle correction, and neighboring return conductors can cancel or enhance the field. Avoid comparing line-current near-field calculations directly with occupational exposure limits without correct averaging and geometrical context.

Two resolved conceptual checkpoints

Question A. What change of B occurs when current direction reverses in a thin wire?

Reasoned answer. The circumferential B direction reverses while magnitude stays unchanged for the same

Question B. Why can a distant paired return conductor decrease magnetic field?

Reasoned answer. Its opposite current produces an opposing far-field contribution; full vector superposition accounts for cancellation.

Next steps

[Unit 6 interactive model](../simulations/model-06.html) · [Assessment](../assessments/unit-06.html) · [Project](../projects/unit-06.html) · [Shared study guidance](../STUDY-GUIDE.html).

Unit 6, lesson 2 — Ampère’s law 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

Ampère’s circuital law is to steady magnetic current what Gauss’s law is to charge: an exact relationship whose convenience depends on symmetry. For a long uniform-current cylinder, a circular Amperian loop gives B proportional to radius inside and inversely proportional outside. For a long solenoid, an axial field approximation is strongest well away from its ends. For a finite transformer core or near a coil terminal, simple one-dimensional magnetic circuits omit leakage and fringing.

Governing relationships and derivation pathway

Magnetostatic integral law: ∮C H·dl = Ifree,enclosed. In free space B=μ0H. Inside a cylinder radius a with uniformly distributed I, Hphi=I rho/(2πa²) for rho

Worked numerical illustration

A solid conductor radius 4 mm carries 10 A uniformly. At rho=2 mm, Hphi=10×0.002/[2π(0.004)²]≈199 A/m; at rho=8 mm Hphi=199 A/m again. Equal numerical values here occur at two special radii, not at every radius.

Independent numerical variant

Inside a uniform solid wire radius 3 mm carrying 6 A, at rho=1.5 mm B≈0.2 mT; at rho=6 mm B also ≈0.2 mT.

Engineering context: Inside versus outside a solid conductor

The B field of a solid uniformly conducting cylinder increases linearly with radius inside and falls as 1/rho outside. Build one continuous piecewise graph and verify that the two branches meet at the conductor surface. The derivative changes there because the source-current distribution changes. If current distribution becomes skin-effect dominated at high frequency, the uniform-current assumption is invalid and the plotted inside branch must be replaced.

Sensitivity and design interpretation

Inside a uniformly conducting solid wire, B grows linearly with radial position; outside it decreases inversely with radius.

Why the model can fail

The surface used to count enclosed current must agree with contour orientation. If material magnetization matters, using B=μ0H everywhere is invalid: account for B=μ0(H+M).

Two resolved conceptual checkpoints

Question A. Where is B maximum for a uniformly current-filled long cylindrical wire?

Reasoned answer. At the surface radius a, where the increasing inside branch meets the decreasing outside branch.

Question B. Can Ampère’s law around a contour enclosing zero net current imply B=0 at every point?

Reasoned answer. No. Only the signed circulation integral is constrained; local B may be substantial.

Next steps

[Unit 6 interactive model](../simulations/model-06.html) · [Assessment](../assessments/unit-06.html) · [Project](../projects/unit-06.html) · [Shared study guidance](../STUDY-GUIDE.html).

Unit 6, lesson 3 — Magnetic materials and field boundaries

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

Magnetic permeability is a constitutive property that relates B to H under specified conditions. Real ferromagnetic cores exhibit nonlinearity, hysteresis and saturation, so a constant relative permeability applies only within a limited range. At an interface, normal B is continuous because magnetic monopoles are absent; tangential H jumps by any free surface current. These relations matter at air gaps, magnetic shields, sensor cores and transformer laminations.

Governing relationships and derivation pathway

∇·B=0; n·(B2-B1)=0. The boundary relation n×(H2-H1)=Kfree (depending on the normal orientation convention). For an ideal linear isotropic material B=μH; magnetic susceptibility gives M=χm H in the linear regime.

Worked numerical illustration

If H=400 A/m and a hypothetical linear core has μr=500, B=μ0 μr H≈0.251 T. This arithmetic cannot predict whether the actual core saturates; the manufacturer’s B–H curve is needed.

Independent numerical variant

For μr=100 and H=1000 A/m, the linear estimate gives B=0.126 T. A nonlinear B–H curve may shift this value substantially.

Engineering context: Air-gap uncertainty

Compute a nominal air-gap B using a constant-μ magnetic circuit. Compare it with a sensitivity range where μr varies by a factor of ten and where the effective gap changes due to mechanical tolerance. If the air gap dominates reluctance, changes in core μ may matter less than gap length. The exercise illustrates why “magnetic material with μr=1000” is not a complete component specification.

Sensitivity and design interpretation

At fixed H, B scales with an assumed linear μ; saturation prevents this extrapolation indefinitely in ferromagnets.

Why the model can fail

Treat hysteresis loops as path-dependent: B at a given H can differ after opposite excitation histories. A high μr number without a frequency and bias specification is inadequate for precision design.

Two resolved conceptual checkpoints

Question A. Which magnetic component cannot jump across an ordinary interface?

Reasoned answer. Normal B remains continuous because div B=0. Tangential H may jump due to surface free current.

Question B. Is a single μr enough to specify hysteresis and saturation?

Reasoned answer. No. Ferromagnetic response depends on bias, frequency, path history and the B–H relation.

Next steps

[Unit 6 interactive model](../simulations/model-06.html) · [Assessment](../assessments/unit-06.html) · [Project](../projects/unit-06.html) · [Shared study guidance](../STUDY-GUIDE.html).

Unit 6, lesson 4 — Vector potential and magnetic flux conservation

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

Because div B=0, it is convenient to introduce a magnetic vector potential A such that B=curl A. The vector potential is not unique: a gradient can be added without changing B. This gauge freedom is useful for computational field solvers, but it means A itself should not be interpreted as a directly measurable unique field. In simple two-dimensional magnetostatics, a single out-of-plane component of A can describe in-plane flux contours.

Governing relationships and derivation pathway

B=∇×A and A_new=A+grad chi produce identical B. In Coulomb gauge div A=0, ∇²A=-μJ for constant μ and suitable steady conditions. Magnetic flux through S is ΦB=∫S B·dA=∮boundary A·dl.

Worked numerical illustration

For a uniform B=0.05 T normal to a 0.01 m² planar loop, Φ=5e-4 Wb. If the loop is rotated so its normal makes 60° to B, the flux becomes 2.5e-4 Wb, highlighting the signed dot product.

Independent numerical variant

A uniform 0.1 T field threading a planar loop of area 20 cm² carries flux .0002 Wb; reversing the normal flips its signed flux.

Engineering context: Vector potential gauge test

Choose a simple A field that produces a uniform magnetic flux density and add the gradient of a smooth scalar χ. Numerically differentiate both A expressions and verify that curl A is unchanged within discretization precision. Then compare their pointwise A values; the difference can be large. This prevents assigning independent physical meaning to an arbitrary vector potential contour while preserving its computational utility for field-solvers.

Sensitivity and design interpretation

Changing A by a pure gradient leaves B unchanged, no matter the scale of that added gradient.

Why the model can fail

Flux through an open surface can be nonzero; flux through a closed surface is always zero in the classical Maxwell theory. A magnetic vector-potential plot is gauge-dependent unless the gauge is specified.

Two resolved conceptual checkpoints

Question A. Does A+grad χ change measurable B?

Reasoned answer. No, because curl of a gradient is zero. This is gauge freedom, provided required regularity holds.

Question B. What must be true of magnetic flux through a closed surface?

Reasoned answer. It is zero in classical electromagnetism: field lines do not begin or end at isolated magnetic charges.

Next steps

[Unit 6 interactive model](../simulations/model-06.html) · [Assessment](../assessments/unit-06.html) · [Project](../projects/unit-06.html) · [Shared study guidance](../STUDY-GUIDE.html).

Unit 6 engineering project: Field of a supply-and-return pair

Problem statement and intended decision

Two long parallel conductors carry +5 A and -5 A, separated by 1 cm. Compute B vectors at a small set of positions using superposition from long-wire expressions. Compare field magnitude to that of a single wire at equal distance. Choose one contour around both wires and verify that enclosed free current is zero despite nonzero local B. Use the field map to justify reducing loop area as one strategy for limiting magnetic coupling. Report where the infinite-wire approximation becomes unreliable.

Detailed investigation plan

Compute the field at a set of points near two parallel opposite currents. Use Biot–Savart superposition in the infinite-wire approximation to draw B vectors, but mark the assumptions that the wires are straight, very long and sufficiently separated from terminations. Use Ampère’s integral law along a loop enclosing only one conductor and then a loop enclosing both. The second has zero net enclosed current, yet local B on its perimeter is not generally zero. This is an accessible example of the danger in interpreting an integral law as a pointwise zero-field prediction.

Turn the analysis into an EMC routing choice. A paired current and close return typically gives a smaller loop area than two widely separated conductors; compare field magnitude at specified observation points. Avoid interpreting the field as an absolute safety rating; near-field exposure depends on time waveform, geometry and applicable measurement standards. Include vector arrows and one independently checked point from direct analytic formulas.

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 6 assessment — Magnetostatic fields and magnetic materials

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 — Magnetic flux density and the Biot–Savart law

Prompt: What change of B occurs when current direction reverses in a thin wire?

Reveal worked reasoning

The circumferential B direction reverses while magnitude stays unchanged for the same |I| in the linear magnetostatic model. The matching lesson and shared study guide provide additional verification questions.

Question 2 — Magnetic flux density and the Biot–Savart law

Prompt: Why can a distant paired return conductor decrease magnetic field?

Reveal worked reasoning

Its opposite current produces an opposing far-field contribution; full vector superposition accounts for cancellation. The matching lesson and shared study guide provide additional verification questions.

Question 3 — Magnetic flux density and the Biot–Savart law numerical and decision challenge

Prompt: Independently derive and check this scenario: A long 8 A wire at 4 cm generates B=40 μT in vacuum, assuming the observation lies outside its thin cross section. Then choose a parameter to change, predict its effect from doubling straight-wire current doubles B; doubling observation radius halves B outside a thin long wire. State which quantities remain fixed and identify one important excluded physical effect.

Reveal worked reasoning and review criteria

Reference calculation: A long 8 A wire at 4 cm generates B=40 μT in vacuum, assuming the observation lies outside its thin cross section.

Sensitivity logic: Doubling straight-wire current doubles B; doubling observation radius halves B outside a thin long wire.

Engineering qualification: A finite conductor needs an end-angle correction, and neighboring return conductors can cancel or enhance the field. Avoid comparing line-current near-field calculations directly with occupational exposure limits without correct averaging and geometrical context. 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 — Ampère’s law and symmetry

Prompt: Where is B maximum for a uniformly current-filled long cylindrical wire?

Reveal worked reasoning

At the surface radius a, where the increasing inside branch meets the decreasing outside branch. The matching lesson and shared study guide provide additional verification questions.

Question 5 — Ampère’s law and symmetry

Prompt: Can Ampère’s law around a contour enclosing zero net current imply B=0 at every point?

Reveal worked reasoning

No. Only the signed circulation integral is constrained; local B may be substantial. The matching lesson and shared study guide provide additional verification questions.

Question 6 — Ampère’s law and symmetry numerical and decision challenge

Prompt: Independently derive and check this scenario: Inside a uniform solid wire radius 3 mm carrying 6 A, at rho=1.5 mm B≈0.2 mT; at rho=6 mm B also ≈0.2 mT. Then choose a parameter to change, predict its effect from inside a uniformly conducting solid wire, B grows linearly with radial position; outside it decreases inversely with radius. State which quantities remain fixed and identify one important excluded physical effect.

Reveal worked reasoning and review criteria

Reference calculation: Inside a uniform solid wire radius 3 mm carrying 6 A, at rho=1.5 mm B≈0.2 mT; at rho=6 mm B also ≈0.2 mT.

Sensitivity logic: Inside a uniformly conducting solid wire, B grows linearly with radial position; outside it decreases inversely with radius.

Engineering qualification: The surface used to count enclosed current must agree with contour orientation. If material magnetization matters, using B=μ0H everywhere is invalid: account for B=μ0(H+M). 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 — Magnetic materials and field boundaries

Prompt: Which magnetic component cannot jump across an ordinary interface?

Reveal worked reasoning

Normal B remains continuous because div B=0. Tangential H may jump due to surface free current. The matching lesson and shared study guide provide additional verification questions.

Question 8 — Magnetic materials and field boundaries

Prompt: Is a single μr enough to specify hysteresis and saturation?

Reveal worked reasoning

No. Ferromagnetic response depends on bias, frequency, path history and the B–H relation. The matching lesson and shared study guide provide additional verification questions.

Question 9 — Magnetic materials and field boundaries numerical and decision challenge

Prompt: Independently derive and check this scenario: For μr=100 and H=1000 A/m, the linear estimate gives B=0.126 T. A nonlinear B–H curve may shift this value substantially. Then choose a parameter to change, predict its effect from at fixed H, B scales with an assumed linear μ; saturation prevents this extrapolation indefinitely in ferromagnets. State which quantities remain fixed and identify one important excluded physical effect.

Reveal worked reasoning and review criteria

Reference calculation: For μr=100 and H=1000 A/m, the linear estimate gives B=0.126 T. A nonlinear B–H curve may shift this value substantially.

Sensitivity logic: At fixed H, B scales with an assumed linear μ; saturation prevents this extrapolation indefinitely in ferromagnets.

Engineering qualification: Treat hysteresis loops as path-dependent: B at a given H can differ after opposite excitation histories. A high μr number without a frequency and bias specification is inadequate for precision design. 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 — Vector potential and magnetic flux conservation

Prompt: Does A+grad χ change measurable B?

Reveal worked reasoning

No, because curl of a gradient is zero. This is gauge freedom, provided required regularity holds. The matching lesson and shared study guide provide additional verification questions.

Question 11 — Vector potential and magnetic flux conservation

Prompt: What must be true of magnetic flux through a closed surface?

Reveal worked reasoning

It is zero in classical electromagnetism: field lines do not begin or end at isolated magnetic charges. The matching lesson and shared study guide provide additional verification questions.

Question 12 — Vector potential and magnetic flux conservation numerical and decision challenge

Prompt: Independently derive and check this scenario: A uniform 0.1 T field threading a planar loop of area 20 cm² carries flux .0002 Wb; reversing the normal flips its signed flux. Then choose a parameter to change, predict its effect from changing A by a pure gradient leaves B unchanged, no matter the scale of that added gradient. State which quantities remain fixed and identify one important excluded physical effect.

Reveal worked reasoning and review criteria

Reference calculation: A uniform 0.1 T field threading a planar loop of area 20 cm² carries flux .0002 Wb; reversing the normal flips its signed flux.

Sensitivity logic: Changing A by a pure gradient leaves B unchanged, no matter the scale of that added gradient.

Engineering qualification: Flux through an open surface can be nonzero; flux through a closed surface is always zero in the classical Maxwell theory. A magnetic vector-potential plot is gauge-dependent unless the gauge is specified. 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.