Learn All 61 finite-plan destinations are authored and independently reviewed against the finite source/action scope: original continuous-charge field derivations, normalization, asymptotic and quadrature error comparisons, and withheld spatial inference. Flux, enclosed-charge, reference-level and potential-consistency work is now included. Conductor/capacitance and energy-inference work is now included. Current dynamics, magnetic sources, induction, oscillating circuits and Maxwell/wave evidence are now included. Final coupled electromechanical storage and spatial/transient calibration dossiers connect all six core categories.
This is the course outline. Open a unit to choose a lesson; every lesson opens on its own page. Explanations and experiments live inside those lesson pages.
Preparation: Algebra, trigonometric components and introductory single-variable differentiation/integration. Density normalization, source parameterization, radians and asymptotic comparisons are explained locally. Prior course completion does not block access.
Course outcome: Derive and test continuous-source electric and magnetic fields, flux and potential constraints, conductor/dielectric storage, circuit differential states, induction and Maxwell-wave accounts. Defend calibrated and unused evidence with bounded error and retained critique across the six finite categories.
All lessons are directly accessible. Reused foundation explanations are identified where relevant; completing a different course does not award this course's evidence. The 61 registered lessons keep separate predictions, captures, practice responses and reflections.
Connect source integrals, field constraints, storage and differential states with independently calibrated evidence, preserved critique and genuinely new transfer.
Before revealing values, compare R=0,1,2 with the same initial speed. Derive the two coupled equations, identify current continuity at release, locate a velocity reversal in the lossless case and reconcile the integrated energy loss with the resistor. State what source or geometric observation would invalidate constant L or uniform applied field.
Fit geometry amplitude and readout offset from exactly two records; withhold the third. Then fit the displayed exponential from exactly two voltage readings and withhold a third. Propagate the two capacitance bounds into clock gain without treating readout offset as stored capacitance. Retain one alternative dielectric profile and one alternative timing explanation that these records alone cannot rule out.
Derive Faraday and LC behavior, interpret AC response and resonance with explicit initial/steady-state boundaries, and connect Maxwell constraints to electromagnetic-wave energy and source inference.
Compare radii 0.5,1,2 m and both ramp signs. Derive the area integral before revealing the ledger, reconstruct circulation from Etheta and circumference, and distinguish nonzero flux from nonzero flux change.
Compare early and late times at fixed drive, then change the drive force. Retain velocity, displacement, kinetic storage and heat. Explain what additional state would be needed if loop inductance were no longer negligible.
Compare n=100,200,400 at the same current, then set current to zero. Record flux per turn, linkage, inductance and both energy accounts. Explain why doubling N at fixed length differs from doubling length at fixed n.
Test both signs of M inside and outside the bound. Retain the determinant before evaluating a chosen state, and explain why the mutual term can be negative while the total admissible energy stays nonnegative.
Compare initial states (1,0),(0,2),(1,2) and (0,0). Predict the quarter-period state before inspecting the phase curve. Explain which initial states have equal energy but different phase and why one observed charge cannot identify both states.
Compare R=0,2,4,6 at the same elapsed times. Retain the branch label, charge, current and energy loss; explain why the critical expression needs a separate repeated-root limit and why the zero-resistance history cannot lose energy.
Compare frequencies below, at and above 2 rad/s for two positive resistances. Retain signed phase, RMS and peak current, real/reactive/apparent power and the power-triangle check. Explain why cancellation of net reactance does not imply zero instantaneous energy in L and C.
Fit only the first two signed reactances and freeze the third-frequency prediction. Compare direct propagation through its observation weights with separately propagated parameter intervals. State what extra evidence would test the assumed frequency and phase calibration.
Compare full and half-radius flux derivatives, and explain how both spanning-surface accounts agree. Change charge amplitude and frequency independently. Predict the field at t=0 before confusing zero charge with zero charging current.
Compare the valid pair with a Faraday-only pair and with reversed magnetic sign. Predict each residual before inspecting the plots. Explain which violation could be interpreted as requiring a current and which cannot be repaired by an electric-current term.
Compare doubled amplitude with doubled duration and derive each scaling before revealing values. Retain both averaging steps, area and momentum idealization. Explain what additional material and geometry evidence would be needed to infer deposited heat or detector response.
Compare transverse and longitudinal gradients at positive, negative and zero coefficient. Compute divergences and curls before assigning sources. Keep electric Gauss, magnetic Gauss, Faraday and Ampère as separate checks rather than claiming that one passing equation validates the whole field.
Use Biot–Savart and Ampere with explicit source closure and symmetry; analyze force, torque and charged motion; compare spatial magnetic data with two independently calibrated hypotheses.
At y=0.25,0.5,1 m compare the exact integral with N and 2N midpoint cells. Retain the signed quadrature error separately from the infinite-wire error. Sketch a possible return route and explain why its field would be an additional physical contribution, not numerical error.
Compare 60°,90°,180° at the same radii, converting each angle to radians before calculating. Then change only outer radius and retain both arc terms. Explain why the zero connector contribution must not be reused for an off-center observer.
Compare separation ratios 0.5,1,1.5 at z/R=0,0.25,0.5. Retain center amplitude, second derivative and normalized departure as separate evidence. Explain why the zero-curvature profile still changes at finite distance.
Change α while comparing one radius inside the inner conductor, one in the empty gap and one outside the return. Predict which observations depend on the inner profile and which depend only on total current. Retain the symmetry statement before extracting B from circulation.
Compare second currents +2,0,−2 A. Record exact circulation, numerical circulation and fields at the rightmost and top points. Explain which symmetry would be required to replace the entire integral by a single measured B times circumference.
Compare both charge signs at the same initial velocity, then set transverse speed to zero. Retain position, curvature direction, period, pitch and energy separately. Differentiate the stated position solution to verify that the initial velocity has not silently changed.
Compare 0°,60°,90°,180° at the same field. Predict the sign of torque from the energy slope before reading the ledger. Explain why zero force, zero torque and a stable energy minimum are three distinct statements.
Compare charge signs and then compare E=0 with nonzero E. Retain electric work and guiding drift rather than attributing the energy gain to magnetic curvature. Substitute the force-free selector velocity into the equations and explain why it differs from the displayed initial state.
Fit both amplitudes and baselines from only distances 1 and 2. Freeze both third-distance predictions and propagate ±0.005 µT calibration bounds. Then compare the unused third observation, retaining any additional observation uncertainty. State an independent measurement needed to separate background from instrument offset.
Connect current-density integrals to circuit balances, solve RC and LR differential models with signed energy, and fit transients with explicit measurement uncertainty and fresh validation.
Keep total current fixed and compare α=0,1,2,3. Record central density and half-radius current before revealing them. Explain why the profile is renormalized and why the partial-current discrepancy is not lost charge.
Compare α=0,1,2 at the same current. Derive the integral before reading the ledger, and reconcile its power with the plotted spatial density. State why this model does not resolve lateral fields near the taper ends.
Compare β=0,1,2 and locate the charge maximum and zero crossing when they occur in the displayed interval. Retain current imbalance and charge separately. Explain the independent ideal-source assumption rather than silently assuming steady state.
Change R2 while holding Rs fixed. Predict which node and source currents change, then retain both node residuals and the independently calculated power residual. Explain why a zero residual checks arithmetic under this topology but does not verify the topology experimentally.
Compare Vi=−2,6,10 V at the same time. Before revealing values, predict current sign, source-work sign and whether field energy rises or falls. Retain the full finite-time energy ledger rather than labeling every source as always delivering energy.
Compare observations just before, exactly at and after a chosen switch. Retain which one-sided convention is used for current. Predict how delaying the switch changes the second initial state without changing the −2 V final asymptote.
Compare i0=−1,2,4 A for the same L. Record initial slope, current at one second, source work, resistor conversion and stored-energy change. Explain why positive dissipative conversion does not require source power to be positive at every instant.
Fit only the first three readings before revealing the fourth. Record the offset-sensitive raw-log mistake and compare it with the difference-ratio solution. Perturb all three calibration readings within their bounds, preserve their shared roles in numerator and denominator, and retain the endpoint interval.
Find two generator pairs with the same product gτ and compare their entire displayed curves. Then apply the independent calibration interval and state whether its premise covers each selected gain. Keep response validation distinct from clock validation.
Compare h=0.5 with four steps and h=0.25 with eight steps, both at u=2. Retain signed voltage error and energy ratio. Then test h=1.5,2,2.5 to separate alternating decay, neutral alternation and growth; do not clip the rejected behavior.
Derive conductor/capacitor relations, analyze dielectric and source-energy constraints, and design a parameter-inference investigation with genuinely withheld evidence.
Compare isolated Q=−1,0,+1 nC with fixed central q=1 nC. Retain inner charge, outer charge, metal potential and cavity field separately. Explain which observations change and why replacing isolation by grounding is a new physical constraint.
Compare d=0.1,0.2,0.5 m at fixed a=1 m. Retain exact capacitance, the chosen planar approximation and its signed relative error. Derive a gap-to-radius criterion for a maximum 5% relative error instead of choosing a visually thin gap.
Compare a=0.005,0.01,0.02 m at fixed ratio 3. Retain capacitance per length and inner field separately. Then vary only the ratio and compare the derivative sensitivity with a small finite change, declaring which geometric quantity is held fixed.
Compare α=0,1,2,4 at fixed voltage. Retain constant D, varying boundary fields and both effective and arithmetic-mean factors. Verify the α=0 limit separately and explain why rearranging the same materials side by side changes the averaging law.
Compare fixed charge and fixed voltage at overlap fractions 0,0.5,1. Retain source work, stored-energy change and external work before inferring force. Explain why the same geometric Cprime does not make force independent of the electrical constraint.
Compare V2=−4,0,+4 V with C1=2 µF. Retain signed total charge, final common voltage and all energy terms. Explain why equal final voltage does not imply equal energy history and why resistance cannot be inferred from the final state alone.
At fixed q2=0, compare q1=−3,0,+3 nC. Retain each node plate charge and reconstruct its specified net charge. Verify energy using both branch terms and half charge dot potential; then change q2 and explain why potential changes at both nodes.
Freeze the third-gap prediction after fitting only the first two records. Compare several stray values without changing S. Propagate reading bounds, then derive the direct third-prediction formula in terms of C1,C2 to avoid treating correlated parameter intervals as independent.
Compare final gap ratios 0.5,1,1.5,2 under both constraints. Retain field, force, stored-energy change and source/external work separately. Verify the exact no-change state and explain why the sign of stored-energy change alone cannot establish force direction under maintained voltage.
Apply flux and potential laws with explicit surfaces/references, and compare direct integration with symmetry and independent path checks.
Compare β=−1,0,1,2 at L=2 m. Retain signed flux, center approximation and its error before inspecting the graph. Explain why a zero net flux through part of the patch would not prove that the local field vanishes.
Change a and b separately, predicting which face pairs change. Retain all six signed terms rather than only the sum. Find two different settings with zero net flux and show that their point fields can differ.
Compare α=0,1,2,3 at radii 0,0.5,1,2 m. Retain density normalization, enclosed charge and field separately. Identify which measurements distinguish density shape and which are identical because they depend only on total charge.
Compare observations below the core radius, inside the material and outside it. Derive both boundary values independently and check continuity. Explain which new premise would be needed before applying the zero-core conclusion to a finite charged tube.
Predict the exact field at each allowed observer before evaluating the midpoint integral. Compare 20,40,100 cells at z=0.5,0.75,1.25,2 m. Retain signed errors and explain why simply increasing cells does not validate an omitted surface-charge or external-field assumption.
Derive both components before comparing the two paths for at least two coefficient pairs. Retain endpoint potentials, signed work and the effect of adding a constant reference offset. At one point choose a vector perpendicular to E and check that its first-order potential change vanishes.
Predict the center field from symmetry, then verify by differentiating the potential. At z=1 m compare h=0.1,0.2,0.4 m and retain signed errors. On paper add the same 5 V offset to both sampled potentials and show the cancellation before discussing independent error bounds.
Compare L=1,3,5 m for the potentials at y=1 and2 m and their difference. Retain the increasing absolute potential separately from the converging difference. Explain why declaring V at infinity zero for the infinite line would contradict the integral.
Predict which coefficient pairs are compatible, then compare two paths and the curl. Retain one rejected pair and state what independent measurement could distinguish calibration inconsistency from a genuinely non-electrostatic setting. Do not report a scalar potential for the rejected state.
Integrate discrete and continuous charge fields with explicit geometry/symmetry, and test convergence and far-field limits against independent evidence.
Predict the north-field sign at x=−0.5,0,+0.5 m with y=1 m before opening the ledger. Retain each displacement and signed component. Reflect the observer across the vertical axis and identify which component is even and which is odd.
Compare α=0,1,2 at fixed Q and L. Predict which way the centroid moves, retain the normalization and partial charge, then verify the full charge integral independently. Explain why a field comparison at fixed λ0 would answer a different question.
At fixed L=1 m, compare y=0.5,1,2,4 m. Record both relative errors before choosing an acceptable tolerance. Repeat with changed L and state whether Q or λ is held fixed; do not combine the two limiting procedures.
Compare α=−0.5,0,+0.5 at fixed y=1 m. Predict the sign of Ex before calculation, retain an odd/even integrand sketch and verify that both total charge and Ey remain fixed. Explain why a centered uniform approximation discards a real component.
Compare half-angles 30°,90°,180° at fixed Q and radius. Retain the radian conversion, λ and center-field vector. Then double R on paper and explain the field scaling under fixed Q versus fixed λ.
At a fixed R, compare center, z/R=0.5,1,2,3 and compute the analytic maximum between supported grid points. Retain derivative signs and distinguish a grid maximum from the true stationary point. Double R and test both the location and magnitude scaling.
At fixed σ, compare increasing R while keeping z fixed. Retain the changing Q as well as the field. Then hold R fixed and compare positive z values; test whether point or plane approximation meets a stated relative-error tolerance.
Keep p and r fixed while reducing d, retaining the required q and both exact fields. Compare signed and magnitude errors separately. Then explain why a distant field measurement of p alone cannot uniquely recover q and d.
At y=1 m, compare N=4,8,16,32 and record signed errors against the exact integral. Repeat at y=0.1 m before asserting a convergence rate. State which discrepancy comes from discretization and which hypothetical independent observation could challenge the source distribution itself.
Freeze the point-model prediction before revealing the second reading. Then explicitly relabel the first two readings as rod-calibration data, infer L and Q times gain, and freeze a third-distance prediction before revealing the record at 3 m. Compare common gains 0.9,1,1.1, propagate the stated reading bounds and propose an independent amplitude reference. Do not count the same second reading as fresh validation of the rod it helped fit.
Keep predictions before comparison, complete raw records and units, explain competing models and test a changed case. Supplied and simulated data are labeled in each lesson. A saved notebook is not proof of experimental technique or correct reasoning.
Evidence to retain: Derive the two state equations and energy derivative before viewing results. Compare lossless, damped and critical branches; preserve a reversal, charge integral and energy account. Explain why the earlier negligible-inductance model cannot be substituted without changing premises.
Boundary: Virtual fixed-field, constant-inductance, lumped model; no actual coil, rail or energy-recovery apparatus. A passive energy matrix and real geometry require independent evidence.
Evidence to retain: Derive the radial dielectric integral, fit exactly two indicated capacitances and withhold a third. Independently fit two zero-baseline transient readings and reserve a third. Propagate hard calibration bounds while distinguishing sensor offset, physical stray and timestamp gain.
Boundary: Prescribed linear dielectric and independently fixed resistance/stray; no unique material identification, confidence level or actual laboratory calibration claim.
Evidence to retain: Integrate a nonuniform flux and reconstruct induced electric circulation. Solve finite-mass motional damping and close drive/storage/conversion energy. Derive solenoid inductance by linkage and volume energy; reject a proposed mutual matrix that fails passivity for all currents.
Boundary: Local imposed fields, ideal thin circuits and long-solenoid approximations with explicit source/return assumptions. No electrical construction, component-rating or magnetic safety claim.
Evidence to retain: Retain both LC initial states, derive all three RLC damping branches and check passive energy. Distinguish peak/RMS and signed reactive power. Fit only two signed reactances, freeze a third frequency and propagate shared-observation bounds directly.
Boundary: Linear ideal element models and separately stipulated phase/frequency calibration. A magnitude-only fit, transient response and steady phasor response are not interchangeable.
Evidence to retain: Reconcile conduction and displacement-current spanning surfaces, evaluate both curl residuals of a proposed plane wave, integrate carrier-averaged pulse energy and audit static source requirements versus magnetic-divergence rejection.
Boundary: Consistent rounded vacuum constants and prescribed local/plane-wave models. No claim of global apparatus realization, exposure, dose, detector performance or observed practical competence.
Evidence to retain: Retain signed segment/arc integrals, omitted versus included return paths, a loop-pair derivative calculation and a normalized enclosed-current profile. Compare analytic and numerical circulation without extracting a constant local field when symmetry is absent.
Boundary: Prescribed thin or infinite source models with rounded educational permeability. No claim that an isolated steady current segment or infinite wire is a complete physical apparatus.
Evidence to retain: Differentiate and integrate the signed helical and crossed-field initial states, check energy and preserve charge sign. Compare loop net force, torque and energy curvature; explain the fixed-current interaction-energy premise.
Boundary: Fictional particles, uniform imposed fields and ideal maintained current; no beam, magnet handling, radiation, collision or source-feedback model.
Evidence to retain: Fit amplitude and baseline independently for finite known-length and infinite-line shapes using only two readings. Freeze a third-distance prediction and propagate calibration plus fresh-observation bounds. Identify what independent evidence would separate instrument offset, background and return-field explanations.
Boundary: A bounded two-hypothesis comparison, not unrestricted inverse-source uniqueness, a measured experiment or a statistical confidence statement.
Evidence to retain: Normalize a radial density, integrate reciprocal tapered area, reconstruct control-volume charge and solve both node equations. Retain a spatial or temporal integral and an independent power/charge check rather than treating one scalar current as a complete field description.
Boundary: Synthetic prescribed profiles, ideal current sources or steady lumped resistors have distinct assumptions. No temperature, real compliance or full three-dimensional material inference.
Evidence to retain: Solve RC and RL equations with nonzero signed initial states; match the switched capacitor boundary and retain one-sided current. Integrate source work, resistor conversion and stored energy. Compare numerical methods at a fixed final time and reject an unphysical stable sign alternation.
Boundary: Ideal linear circuit elements and bounded virtual histories; no live source, capacitor handling, energized inductor switching or component-rating guidance.
Evidence to retain: Fit three unknown exponential parameters using only three exact-time readings, freeze a fourth-record prediction, propagate bounded errors through shared differences, and distinguish response validation from independent clock calibration.
Boundary: Deterministic synthetic calibration packets, not confidence intervals, authenticated independent data collection or unique physical mechanism identification.
Evidence to retain: Retain cavity/outer charge balances and isolation versus grounding. Derive spherical/coaxial voltage integrals with explicit geometry limits, then solve a floating-node charge matrix and reconcile branch versus charge-dot-potential energies.
Boundary: Ideal conductors and vacuum geometry, with stated concentric/long-source symmetries; no finite-apparatus or equilibrium claim outside those conditions.
Evidence to retain: Integrate reciprocal graded permittivity and local energy, compare fixed-charge/fixed-voltage insertion and gap change, retain source/external work and account for the loss during charge sharing.
Boundary: Stipulated linear lossless dielectrics or positive-resistor equilibration are different models. No insertion, charged-device handling, fringe resolution, pull-in or breakdown prediction.
Evidence to retain: Fit only two exact-gap readings, freeze a third-gap prediction, retain correlated parameter-error bounds and compare gap-zero/gain/material alternatives. Propose independent geometry or amplitude evidence rather than identifying a real material from one effective capacitance.
Boundary: Synthetic calibrated-gain observations and deterministic reading bounds; no confidence level or unrestricted source/material uniqueness.
Evidence to retain: Retain signed patch integration, all six box faces, a normalized radial charge integral and all annular branches. Explain why zero net flux does not determine a local field, and which full-source symmetries justify factoring a Gaussian integral.
Boundary: Specified mathematical charge distributions and boundary conditions, not real source reconstruction or conducting equilibrium.
Evidence to retain: Compare normalized sphere versus shell interior/exterior predictions. Evaluate two latitude grids against an independent shell benchmark, then derive a ring field from its scalar potential and quantify finite-difference error.
Boundary: Ideal smooth source models with excluded charged-surface self-field; quadrature and derivative errors are not experimental confidence intervals.
Evidence to retain: Close two oriented work paths against endpoint potentials, compare a finite-rod reference with an infinite-line difference, and preserve a rejected nonconservative candidate without inventing a unique endpoint potential.
Boundary: Static electrostatics on the stated simply connected region; induction or inconsistent sensors require new premises rather than a silent scalar fit.
Evidence to retain: Retain source-to-observer vectors, signed components and a negative test-force example. Normalize a nonuniform rod, verify its partial charge/moments, and derive how density asymmetry defeats support-only cancellation. Parameterize arc length and declare fixed Q versus fixed density.
Boundary: Prescribed nonconducting static source distributions in vacuum and a rounded educational Coulomb coefficient, not equilibrium conductors or measured sources.
Evidence to retain: Compare exact line/ring/disk/dipole fields with explicit limiting models, derive a ring stationary point, and retain signed relative errors at two geometries. Run N and 2N midpoint grids against an independent analytic integral and explain a near-source resolution failure.
Boundary: Point and continuous-source models have excluded singular domains; numerical convergence does not certify physical calibration or absent effects.
Evidence to retain: Fit the point-source amplitude only to the first field and freeze its second-distance prediction. Then explicitly use both readings to calibrate the unknown-length rod, freeze a fresh third-distance prediction and reveal that third record. Propagate bounded reading errors through the ratio inversion; separate length from charge/gain and calibration from validation.
Boundary: A restricted synthetic family and deterministic compatible bounds, not a unique real source reconstruction or probabilistic confidence claim.
Open the retained response and project workspaces. Practice, saved work and independently reviewed proficiency remain separate.
These 36 course-specific public tasks are practice, not a secure examination. Each retains up to three immutable written responses, known feedback exposure before each response, and human rubric critiques. Drafts remain editable. No text-length check or self-reported review certifies correctness, independence, laboratory skill or mastery.
A rod of length L=3 m has total charge 6 nC and density λ=λ0(1+s/L), s from 0 to L. Derive λ0, first-half charge and centroid. Separately a uniform rod with half-length 2 m and Q=4 nC is observed at perpendicular distance 1 m; use K=9e9 to compare exact E=KQ/[y sqrt(Lhalf²+y²)] with its point and infinite-line alternatives. State why agreement between approximations is not validation.
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.
A solid nonconducting sphere of radius R has total Q and density proportional to r². Normalize ρ(r), derive enclosed charge and E inside, and find V(0) relative to infinity. Explain why flux alone would not give local field on an arbitrary surface without symmetry. Use K=1/(4pi epsilon0).
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.
An ideal parallel capacitor starts C=2 microF,V=10 V. A quasistatic dielectric change doubles C. Compare isolated fixed charge and ideal fixed voltage: final V or Q, stored-energy change, source work and external mechanical work. Assume reversible dielectric response and no loss; define positive external work into the capacitor system.
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.
A zero-time reference and later records at t=0,1,2 s are 9,5,3 V. Fit y=A exp(−t/tau)+B, freeze the t=3 prediction, and explain how common vertical offset and timestamp gain affect identification. If every input is independently bounded ±.02 V, bound the ratio r=(y1−y2)/(y0−y1) using the shared middle reading rather than unrelated difference errors.
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.
In a fictional unit system m=1,q=−1,Bz=1, initial velocity (2,0,3), find position at t=pi/2 from the origin. Give radius, pitch per positive elapsed period and kinetic energy change. Separately derive the center field of a closed sector loop with inner radius1 m, outer radius2 m, anglepi/2 and current2 A, inner arc counterclockwise. Use μ0=4pi e−7 and explain the radial-segment contribution.
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.
A proposed vacuum wave has Ey=E0 cos(kx−omega t), Bz=2E0/c cos(kx−omega t), omega=ck/2. Test both curl laws. Separately a valid plane pulse has cycle-averaged intensity I0 sin²(pi t/T), uniform area A and normal incidence. Integrate its energy and compare absorbed/reflected momentum. Does passing one curl equation or matching E/B by itself validate a complete field?
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.
For unit mass, inductance and coupling with R=2, v0=3 and I0=0, derive v and I at t=1, retained energy, resistor conversion and net transported charge. Compare the R=0 state at t=pi. Explain why instantaneous I=v/R is not the model being solved.
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.
Use fi=1/ln[2i/(1+i)]. Two capacitance indications at ratios2 and3 are y2=1.5f2−.2 and y3=1.5f3−.2 nF. Infer amplitude and readout offset, then freeze ratio4. A separate physical stray.6 nF and exact resistance3 Mohm accompany ratio4; a zero-baseline transient has displayed time constant1.1 times the physical value. Infer gain and bound it if each first capacitance indication has independent hard error±.01 nF while displayed tau is exact. Identify which new record tests geometry and which tests time response.
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.
For the taught nonuniform Bz=(2+beta t)[1+(r/ell)²],ell1 m,loopR1 m,beta−2 T/s, find flux derivative, counterclockwise emf and azimuthal E. Explain why a scalar electrostatic potential cannot reproduce nonzero circulation. Separately use the unit-parameter moving rod with drive1 N at t=ln2 s: find speed, displacement, drive work, kinetic storage and resistor conversion.
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.
For the taught long solenoid with length2 m,area.01 m²,n100 per metre,I3 A, derive L and stored energy using both linkage and field volume. For a separate reciprocal pair L1=1 H,L2=4 H,M=2 H,I1=1 A,I2=−.5 A, compute energy and explain the ideal equality boundary. Would M=3 H be admissible merely because the trial I1=I2=1 A gives positive formal energy?
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.
For L1 H,C.25 F,q0=1 C,I0=2 A, find the exact quarter-period charge/current and both energies. Compare with initial state q0=−1,I0=−2. Separately the same L/C has R4 ohm,q0=1,I0=0: derive the repeated-root solution and find charge,current,stored energy and total resistor conversion at t1 s.
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.
With source6 V RMS,L1 H,C.25 F,R3 ohm and omega1 rad/s, find X,impedance magnitude,RMS current,phase,real/reactive/apparent powers. Independently an unknown L/C pair gives calibrated imaginary impedances X1=−3 ohm at1 rad/s and X2=0 at2 rad/s. Fit L,C and predict the unused3 rad/s reactance. Each calibration error is bounded.03 ohm: derive the direct prediction bound and distinguish it from a probabilistic confidence interval.
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.
For Q(t)=3 nC sin(2t) on the taught radius1 m capacitor, find lead current,full-gap displacement current,half-radius enclosed displacement current and B at half radius at t0. Does Q0 at that instant imply no magnetic field? Separately test the plane-wave candidate a=.5,b=2,E0=2 V/m,k=pi per m,c3e8. Which curl constraint passes, and what sinusoidal electric-current coefficient would the other require under ordinary Maxwell theory?
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.
A locally plane vacuum pulse has carrier amplitude envelope3 sin(pi t/T) V/m over T4 microseconds,with intercepted area.01 m². Derive fluence and intercepted energy using the consistent teaching epsilon0 and c, then give absorbing and perfect-reflecting momentum transfers. Separately static E=(−x,0,0) V/m and B=(0,−1e−6 x,0) T have x in metres: find required charge/current density. What changes if the same magnetic gradient is instead in the x component?
Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.
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Use the rounded mu0=4pi×10⁻7. A straight x-directed segment has half-length2 m,current2 A and observer(0,1,0) m. Derive its field vector and compare the infinite-wire approximation. Separately a closed sector carries3 A counterclockwise on inner radius1 m and clockwise on outer radius2 m, with sector angle60°. Find the center field and explain the radial connector contribution. Why are the two source-completeness claims different?
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For the taught coaxial source use alpha1 and radius.5 m: derive enclosed current and field. At radius sqrt6 m inside the return annulus, find them again. Separately two offset infinite wires carry+2 and−2 A; explain why zero circulation on an enclosing centered circle does not force zero local field, and calculate its field at(0,1) m.
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For equal coaxial loops of radius R and same current I, derive why center first derivative vanishes for every separation d and find the d at which the second derivative vanishes. Does this establish an exactly uniform field? Separately the taught current loop has moment.6 A m² in B=2 T at120°. Find angular torque toward increasing angle, effective orientation energy and quasistatic work from alignment. Compare aligned and anti-aligned energy curvature.
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For the fictional m1 kg,q−1 C particle in B+1 T along z, start with velocity(2,0,2) m/s and no E. Find position at pi/2 s, radius,pitch and magnetic work. In a separate rest-start experiment add E+1 V/m along x with the same B and charge; find position and velocity at pi s and the electric work. Which initial velocity would instead be force-free?
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At exact distances1 and2 m, indicated fields are .1+.6/sqrt2 and .1+.6/(2sqrt5) microtesla. Fit both A f(y)+B0 with f=1/[y sqrt(L²+y²)] with L=1 m and C/y+D. The finite half-length1 m is independently stipulated. Freeze each third-distance prediction, then compare the fresh indicated value .1+.6/(3sqrt10). Each calibration record and the fresh record has independent error bound .005 microtesla. Derive each prediction bound using its direct weights and decide whether the infinite-line residual exceeds the combined bound. Can this identify the physical baseline uniquely?
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A radius 1 mm disk carries 2 A with J=J0[1+3(r/R)²]. Derive J0 and the current within half radius. In a separate slender element, rho=2e−8 ohm m, L=1 m, A0=1 mm² and radius r=r0(1+2x/L); at 3 A derive resistance and power. Finally integrate prescribed inflow 3 mA and outflow (1+2t) mA from initially zero charge to t=3 s. Explain why matching instantaneous flows is not zero stored charge.
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In the taught two-node network set Rs=R2=1 ohm. Solve both node potentials and all source/bridge/ground currents, then reconcile delivered and resistor powers. Separately the taught RC element switches from 6 to −2 V at ts=ln 2 s, having started uncharged. Find its voltage immediately at the switch and at t=2 ln 2 s, and its current jump. State the one-sided convention at the switching instant.
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For the taught RC element use Vi=10 V and t=ln 2 s. Find voltage, signed current, source work, stored-energy change and resistor conversion. Separately take L=2 H,R=2 ohm,source 4 V,i0=4 A at the same elapsed time. Find current, source work, resistor conversion and stored-energy change; retain the cross term when integrating i².
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For y=A exp(−t/tau)+B, exact-time readings at 0,1,2 s are 5,3,2 V. Fit all three parameters and freeze y at 3 s. Each calibration reading has independent bound ±.01 V: derive the extreme ratio and tau endpoints using the shared middle reading, not independent numerator/denominator errors. Separately a clock-scaled zero-offset response fits displayed T=2 s; independent clock evidence says g in [.9,1.1] with displayed time s=g t. Bound physical tau. Can another voltage record alone identify g?
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For normalized discharge vprime=−v,v(0)=1, compare explicit and backward Euler at normalized final time 2 using h=.5 with four steps and h=.25 with eight steps. Give all four voltages and compare with exp(−2). Derive the explicit amplification condition for magnitude decay, then explain why h=1.5 is stable in magnitude yet not a faithful positive RC decay.
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Use K=9×10⁹. A central charge 2 nC lies within the taught shell of inner radius1 m, outer radius2 m and isolated net shell charge−1 nC. Find both surface charges, conductor potential and field at r=.5 and3 m. What changes if it is grounded with no external sources? Separately derive exact spherical capacitance for inner radius.5 m and outer radius1 m; compare the inner-area planar approximation. Explain why doubling both coaxial radii leaves capacitance per length fixed but changes surface field at fixed voltage.
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For epsilon(z)=epsilon0(1+3z/d), derive the effective relative factor for through-gap capacitance and compare it with the arithmetic mean. In the taught lateral model C0=100 pF,ell=.1 m,kappa4,initialV10 V, find C,V and pull at half overlap under fixed initial charge. Under maintained10 V instead, find source work, stored-energy change and external work from zero overlap. Explain the role of edge fields in this bulk energy estimate.
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Capacitors C1=1 µF,C2=3 µF start at6 and2 V and equilibrate through a positive finite resistor in isolation. Find final voltage, initial/final energy and heat. Separately use the taught floating-node network with q1=0,q2=−2 nC: find both potentials, reconstruct node1 net charge and compute stored energy from half q dot V. Is zero net charge on node1 enough to call all its capacitors uncharged?
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At exact gaps1 and2 mm, capacitance readings are35 and20 pF. Fit S and Cs in C=S/d+Cs and freeze the prediction at3 mm. Each calibration reading independently has error bound.1 pF. Derive bounds on S and Cs, then express the third prediction directly in C1,C2 and derive its tight observation-error bound. Why is combining the separate S/Cs intervals as independent unnecessarily loose, and what could a third residual test?
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On a patch z = 0, 0≤x≤2 m, 0≤y≤1 m with +z normal, let E=(5,0,2+2x²) in consistent SI. Find exact flux, center-value approximation and its signed error. Separately in the taught box use E=(2x+3,−y−2,1−z): list all six outward fluxes and total flux. Does its zero total imply zero field at the origin?
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A radius 1 m insulating sphere carries 3 nC with rho=rho0(1+2r²/R²). Normalize rho0 and find Qenc/E at r=.5 m, then E at2 m. A distinct infinite annular source has rho 1 nC/m³, core radius 1 m and outer radius 2 m: find E at r=.5,1.5,3 m. Finally compare exact fields of a uniform thin shell Q2 nC,R1 m at z=.5 and2 m with a nonzero coarse interior quadrature residue.
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For V=x²−y²+2xy+7 V in consistent SI, find E at(1,2), DeltaV from origin and electric work on a−2 nC test charge. Check both axis-aligned paths. A separate ring Q2 nC,R2 m has infinity reference: find center potential and field. For an ideal infinite line with lambda1 nC/m, find V(.5 m)−V(1 m) and explain why V(infinity)=0 is not a finite reference for that source.
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Candidate E=(3y,x) in consistent SI is claimed electrostatic on the whole rectangular region from(0,0) to(1,2). Compute east-then-north and north-then-east work per charge, the loop difference and curl. Can an additive potential reference fix the discrepancy? Change the candidate to E=(3y,3x): give one valid scalar potential and endpoint difference.
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Use K=9×10⁹ N m²/C². The first packet has +1 nC at (−1,0) m and −1 nC at (1,0) m; find E at (0,2) m and the force on a separate −2 nC test charge. A different 2 m rod has Q=4 nC and λ=λ0(1+s/L); derive λ0, the first-half charge and centroid. Finally explain why a linear density gradient on a symmetric support need not have zero transverse field.
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For a uniform rod Q=2 nC, half-length1 m and observer distance1 m, compare exact, point and infinite-line fields. For a fixed-Q semicircle with Q=2 nC and radius2 m, find the center field vector. For a uniform ring with Q=2 nC and radius2 m, derive the positive axial maximum location/value. State why a disk integral needs dq=2πσr dr and what changes if radius increases at fixed σ.
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For fixed p=1 nC m, half-separation0.5 m and axial observer2 m, compute q, exact field, leading dipole field and its signed relative error. Explain why equatorial field points opposite p. Separately a2 m source rod is discretized into20 equal midpoint cells and observed0.5 m from its midpoint: find Δs/y. Can agreement of20 and40 cells validate the dipole approximation or the source density?
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A uniform rod of unknown half-length is observed at perpendicular distances1 and2 m. Indicated fields are18/√2 and9/√5 N/C. Infer length from their ratio, fit the point-source alternative at the first location and predict its second field. If both readings instead have a common gain1.1, what changes? After using both readings to calibrate the rod, predict its fresh field at distance3 m and identify which record remains validation for that fitted rod. With independent errors bounded.02 N/C, write the two ratio endpoints and the inverse that maps them to a length interval. Is the result a confidence interval or unique physical source identification?
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Reviewer identity and role are declarations, not authenticated credentials. Unsaved review fields below are lost on navigation. Review every criterion with an exact quotation, calculation or evidence citation and a next step.
Retain the six prior category reports and defend all eight exit packets, including the changed coupled-state and dielectric/transient dossiers. The finite SC08 course has 61 authored destinations; real practical technique, learner mastery and institutional credit are separate evidence gates.
Use changed critical rod v0=3 at t=1, and changed dielectric amplitude1.5/readout offset−.2/physical stray.6/resistance3. Draw system and data-role boundaries before calculating. Reserve a third geometry and a third voltage record.
Evidence to retain: Immutable first report with exact problem data, derivations, source assumptions and frozen unused predictions.
Solve all eight exit packets and cite one actual investigation from each core category. Reconcile charge, field constraints, storage and source work where applicable; derive the original differential or integral relation rather than reuse an unsupported formula.
Evidence to retain: Retained free responses and reasoning, independent residual checks and one rejected alternative model per final chain.
Obtain criterion-linked critique of calculus, initial states, calibration independence and uncertainty. Preserve original work and revise each identified weakness; record unresolved model or source limits without inventing measurements.
Evidence to retain: Actual reviewer identity/criteria, immutable first report, critique and revised report.
A reviewer supplies an actually unfamiliar packet spanning at least three categories, changing one boundary or source premise and one measurement constraint. Declare prior exposure and assistance before independent response.
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Retain raw-data identifiers, provenance, units, prior predictions, methods, calculations, competing explanations and safety boundaries. State explicitly when data are simulated, supplied or genuinely observed. This saved text cannot verify a real experiment; do not invent measurements or completed supervised work. Maximum 12,000 characters; retain larger evidence files separately and cite them.
A nonempty report is only a saved submission. First and revised reports remain separate and cannot be overwritten. A critique of the first report is required before retaining one revision.
The reviewer must supply genuinely new evidence, not simply repeat public numbers. Preserve first work, criterion critique, revision, assigned packet and response across backups and later curriculum versions. Public tests and synthetic fixtures are not actual learner achievement.
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