SCIENCE · INVESTIGATE AND EXPLAIN

Reconstruct a uniform field from a voltage survey

A local uniform-field region can be described by a linear potential V=ax+3y+C in SI units. Three probes occupy (0,0), (2,0) and (0,1) m. Voltage differences along horizontal and vertical baselines determine the components without calculus: electric work gives Ex=−[V(2,0)−V(0,0)]/(2 m) and Ey=−[V(0,1)−V(0,0)]/(1 m). The minus sign says field points toward decreasing potential.

Learning outcome: Change only C first and retain the unchanged field. Then change a and draw the new normal and equipotential direction with common spatial scales. Compare independent endpoint uncertainty with a common offset, and propose a fourth probe location that tests the uniform-field assumption.

Starting recap

Use signed charge, common coordinate axes, SI units, vector components and algebraic ratios. Separate a source field from probe force and a fitted parameter from independently tested evidence.

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

Try

At a=2 V/m and C=1 V, the three voltages are 1,5,4 V. They imply Ex=−2 and Ey=−3 N/C, magnitude √13. A displacement (3,−2) m gives ΔV=6−6=0 and zero electrostatic work for any test charge.

Open the experiment

Predict

Three collinear voltage probes cannot generally determine:

Your prediction and reason

Experiment

Change only C first and retain the unchanged field. Then change a and draw the new normal and equipotential direction with common spatial scales. Compare independent endpoint uncertainty with a common offset, and propose a fourth probe location that tests the uniform-field assumption.

Infer a uniform field from three noncollinear voltages

Worked teaching model · synthetic values · Supplied local planar potential V=a x+3 y+b in SI units. Probes at (0,0),(2,0),(0,1) m have exact coordinates. Compare a common voltage offset with independent ±.1 V per-reading bounds. This is not a global isolated-source potential.

x slope of potential: 2 V/m Common potential offset: 1 V

Reset keeps your prediction, captures and practice history.

Parallel equipotentials on common spatial scales

Spatial axes use the same pixels per physical unit, preserving geometry. 1: V minus offset = -3 V (solid).2: V minus offset = 0 V (dashed).3: V minus offset = 3 V (dotted). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Parallel equipotentials on common spatial scales

V minus offset = -3 V

x (m)	y (m)
-2	0.333333
2	-2.33333

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.

V minus offset = 0 V

x (m)	y (m)
-2	1.33333
2	-1.33333

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.

V minus offset = 3 V

x (m)	y (m)
-2	2.33333
2	-0.333333

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.

Voltage differences determine components

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: common-offset voltage map (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Voltage differences determine components

common-offset voltage map

x (m) along y=0	potential (V)
0	1
2	5

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
Origin voltage	1	V
x-probe voltage at (2,0)	5	V
y-probe voltage at (0,1)	4	V
Recovered electric field x	-2	N/C
Recovered electric field y	-3	N/C
Field magnitude	3.60555	N/C
Worst-case Ex error from independent voltage bounds	0.1	N/C
Worst-case Ey error from independent voltage bounds	0.2	N/C
Potential change along displacement (3,−a)	0	V

A common offset cancels from differences, but independent probe errors do not.

Uniform-field work gives Ex=−[V(2,0)−V(0,0)]/2 and Ey=−[V(0,1)−V(0,0)]. The displacement (3,−a) has zero dot product with the field, so it lies along an equipotential. Independent ±.1 V endpoint errors yield difference bound ±.2 V; divide by each baseline. Collinear probes alone would not identify the perpendicular field component.

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

A common offset cancels from differences, but independent probe errors do not.

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 source and measurement boundary

Derive the quantitative comparison

A common additive C changes every voltage equally and disappears from differences. Independent bounded readout errors behave differently. If each endpoint voltage can differ by ±.1 V, their difference can be wrong by ±.2 V. The resulting worst-case field-component errors are .1 N/C along the 2 m baseline and .2 N/C along the 1 m baseline. These are deterministic bounds, not statistical confidence intervals; a common gain error would need a separate calibration.

Test an alternative before claiming identification

An equipotential displacement satisfies aΔx+3Δy=0. The direction (3,−a) is therefore tangent to every straight equipotential in this uniform model. The field (−a,−3) is perpendicular to that direction and does no work along it. Collinear probes cannot identify a perpendicular component: a whole family of different fields could agree along one line. Three points identify this affine model, but they do not establish that the potential stays affine elsewhere; a withheld fourth noncollinear location tests that extension.

Worked original case

At a=2 V/m and C=1 V, the three voltages are 1,5,4 V. They imply Ex=−2 and Ey=−3 N/C, magnitude √13. A displacement (3,−2) m gives ΔV=6−6=0 and zero electrostatic work for any test charge.

Investigation

Changed-case practice

Try the changed case before feedback

Voltages at (0,0) and (0,2) m are 1 and 9 V in a uniform region. Find Ey in N/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 evidence boundary

Original teaching, numerical cases and diagrams checked 30 October 2026 against the relevant primary textbook explanation. No textbook exercise, image or passage is reproduced. These are bounded synthetic records, not observed apparatus. No charging, wiring, capacitor handling, medical application or safety certification is requested. Preserve first predictions, contrasting settings and critiques in the course evidence workspace; public feedback is not independent mastery or performed laboratory time.

Apply

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

Which independent observation could reject this model rather than merely refit its parameters?

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: End an electric trajectory at its first boundary event Next: Separate a plate capacitance from a stray contribution

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.