SCIENCE · INVESTIGATE AND EXPLAIN

Build a signed source-to-observer field kernel

A point-source field is a vector-valued function of the observer position. Keep the source coordinate separate from the observation coordinate: the displacement P−rs points from source to observer, and its magnitude enters the inverse-square law. Combining the unit vector with the distance gives Kq(P−rs)/|P−rs|³. This representation makes sign and geometry auditable rather than hiding them in an arrow sketch.

Learning outcome: 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.

Starting recap

Review signed vector components, definite integrals, radians, density units and elementary derivatives. Distinguish observer coordinates from source integration coordinates.

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Try

At P=(0,1) m, each distance is √2 m. The positive source contributes (9,9)/(2√2) N/C and the negative source contributes (9,−9)/(2√2) N/C. Thus E=(9/√2,0) N/C. A separate −2 nC test charge feels Fx=−18×10⁻⁹/√2 N, opposite the field.

Open the experiment

Predict

At P=(0,1) m, which net field follows from the stated opposite sources?

Your prediction and reason

Experiment

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.

Construct source-to-observer displacement vectors

Worked teaching model · synthetic values · Prescribed +1 nC at (−1,0) m and −1 nC at (1,0) m in vacuum. Observer always lies above the source line. Use rounded K=9×10^9 N m²/C². Field excludes a source acting on itself; force is illustrated for a separate −2 nC test charge without source rearrangement.

Observer east position: 0.5 m Observer north position: 1 m

Reset keeps your prediction, captures and practice history.

Source-to-observer displacement geometry

Spatial axes use the same pixels per physical unit, preserving geometry. 1: positive source to observer displacement (solid).2: negative source to observer displacement, not field direction (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Source-to-observer displacement geometry

positive source to observer displacement

east position (m)	north position (m)
-1	0
0.5	1

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.

negative source to observer displacement, not field direction

east position (m)	north position (m)
1	0
0.5	1

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
Positive-source east field	2.304139	N/C
Positive-source north field	1.536093	N/C
Negative-source east field	3.219938	N/C
Negative-source north field	-6.439876	N/C
Total east field	5.524077	N/C
Total north field	-4.903783	N/C
Field magnitude	7.386644	N/C
Negative test-charge east force	-1.10482e-8	N
Negative test-charge north force	9.80757e-9	N

The signed source charge multiplies the source-to-observer displacement divided by its cubed length.

For each source use E=Kq(P−rs)/|P−rs|³. The displacement vector always points from source to observer; a negative q reverses the resulting field, not the geometry. Sum components before taking a magnitude. F=qtest E changes sign for a negative test charge; the source field itself does not.

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

The signed source charge multiplies the source-to-observer displacement divided by its cubed length.

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 observer

Derive before approximating

The supplied pair is +1 nC at (−1,0) m and −1 nC at (1,0) m. At an observer (x,y), form each displacement, cube its length, multiply by its signed charge, then add east and north components. Only after addition should you take a magnitude. The source-to-observer displacement for the negative charge still points toward the observer, but its field contribution points opposite that displacement.

Check evidence and the domain

Every model in this first category uses the explicitly rounded educational coefficient K=9×10⁹ N m²/C² in vacuum. It is not declared an exact fundamental constant. The prescribed sources do not rearrange when a separate small test charge is introduced; otherwise the field problem would change. The displayed observer stays away from source singularities. No self-force, material breakdown or real apparatus is modeled.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

At P=(0,2) m, retain the same sources. Find the east field 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 boundaries

Original derivations and numerical packets checked 30 October 2026 against the primary explanation. The electrostatic field construction also uses continuous source integration. All records are synthetic; no electrical apparatus, high-voltage handling or practical certification is requested. Public feedback is not independent mastery; retain first predictions, alternative models and criterion-linked critique in the separate course evidence 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.

Next: Normalize charge before comparing density moments

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