SCIENCE · INVESTIGATE AND EXPLAIN

Reject an incompatible scalar-potential claim

A proposed vector field must pass a consistency check before it is called electrostatic. Consider candidate E=(a y,b x) on a simply connected planar region. The coefficients may be supplied by separate fits. Even if each component fits its own data, the pair need not be the negative gradient of one scalar potential.

Learning outcome: 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.

Starting recap

Review oriented normals, definite integrals, source volume measures and partial derivatives as one-coordinate rates. Potential differences use an explicit reference and field work uses a signed line integral.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

At a=1,b=2, the two path works are 4 and2 J/C, so the oriented loop gives2 J/C and no static scalar potential fits both. At a=b=2, both paths give4 J/C, V=−2xy+C and the origin-to-endpoint potential difference is−4 V. Zero loop work here follows from the full smooth field form, not one isolated accidental numerical cancellation.

Open the experiment

Predict

For a=1,b=2, which conclusion follows under the stated static simply connected assumptions?

Your prediction and reason

Experiment

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.

Reject a path-dependent electrostatic scalar claim

Worked teaching model · synthetic values · Candidate planar field E=(a y,b x) in consistent SI. Compare (0,0) to (1,2) along east-then-north and north-then-east paths. Only a=b permits a global electrostatic potential on this simply connected region. Rejected states are candidate-data inconsistencies, not accepted static electrostatic fields; time-varying induction is not modeled here.

Coefficient in east field a y: 1 N/C/m Coefficient in north field b x: 2 N/C/m

Reset keeps your prediction, captures and practice history.

Candidate field tested on two paths

Spatial axes use the same pixels per physical unit, preserving geometry. 1: first path, forward (solid).2: second path, forward before loop reversal (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Candidate field tested on two paths

first path, forward

east coordinate (m)	north coordinate (m)
0	0
1	0
1	2

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

second path, forward before loop reversal

east coordinate (m)	north coordinate (m)
0	0
0	2
1	2

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
East-then-north candidate work per charge	4	J/C
North-then-east candidate work per charge	2	J/C
First forward then second backward loop work per charge	2	J/C
Candidate curl z component	1	N/C/m
Compatible with a static scalar potential	0	flag

Nonzero loop work rejects a static scalar-potential representation under the stated assumptions.

The first path accumulates 2b and the second 2a joules per coulomb. Their oriented loop difference is 2(b−a), equal to curl b−a times rectangle area2. If a=b, V=−axy differentiates to the candidate field; otherwise no single-valued electrostatic potential on this simply connected region can reproduce both paths. Measurement inconsistency or a time-varying magnetic setting would require new premises, not an arbitrary choice of one path as the potential.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Nonzero loop work rejects a static scalar-potential representation under the stated assumptions.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Define the geometry and reference

Derive the integral or gradient

From (0,0) to (1,2), the east-then-north path gives work per charge 2b, while north-then-east gives 2a. Following the first forward and the second backward gives oriented loop work 2(b−a). The local curl component is ∂Ey/∂x−∂Ex/∂y=b−a, consistent with the loop area 2 m².

Check the inference boundary

If a=b, V=−axy plus a constant differentiates to the candidate field and both paths give the same potential difference. If a differs from b, a single-valued static electrostatic scalar is incompatible with these premises. The workbench rejects that claim and withholds a unique endpoint potential. Instrument inconsistency or time-varying magnetic induction would require new evidence and a different model, not silently selecting one path as the answer.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For a=3,b=1, find first-forward/second-backward loop work per charge in J/C.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original calculations checked 30 October 2026 against the primary explanation. All source distributions, fields and readings are hypothetical. This is not a physical high-voltage or electrical handling activity, and public numerical feedback does not establish independent mastery. Preserve first predictions, critique, revision and newly assigned work in the separate retained course workspace.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which geometry, calibration or source assumption could fail while an exact arithmetic fit remains possible?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Choose reference levels before taking an infinite-source limit Next: Separate conductor equilibrium from grounding

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.