SCIENCE · INVESTIGATE AND EXPLAIN

Quantify finite-line end corrections

A source may look like a point from far away and like a long line from close to its midpoint, but those approximations have different domains. The uniform rod extends from −L to L and carries total charge Q. Its perpendicular-bisector observer is at y>0. Pairing elements at opposite source coordinates cancels horizontal components before evaluating the remaining integral.

Learning outcome: 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.

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

With Q=2 nC, L=1 m and y=2 m, the exact field is 9/√5≈4.024922 N/C. The point comparison is 4.5 N/C, an overestimate √5/2−1≈0.118034. The infinite-line comparison is 9 N/C, an overestimate √5−1≈1.236068.

Open the experiment

Predict

Which relative overestimate belongs to the point approximation at L=1 m and y=2 m?

Your prediction and reason

Experiment

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.

Quantify finite-line versus point and infinite-line limits

Worked teaching model · synthetic values · Uniform rod from −L to L m has fixed total charge 2 nC. Observe its perpendicular bisector at positive y. K=9×10^9. Infinite-line comparison uses this rod’s λ=Q/(2L), while the point comparison uses its Q. Approximation errors are deterministic, not measurement uncertainty.

Rod half-length: 1 m Perpendicular observation distance: 2 m

Reset keeps your prediction, captures and practice history.

Finite rod bisector field

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

Equivalent numerical coordinates for Finite rod bisector field

finite-source integral

observer distance (m)	field (N/C)
0.25	69.85026
0.296875	58.12427
0.34375	49.51959
0.390625	42.92155
0.4375	37.69333
0.484375	33.44447
0.53125	29.92204
0.578125	26.95477
0.625	24.42235
0.671875	22.23761
0.71875	20.33569
0.765625	18.66723
0.8125	17.19391
0.859375	15.88544
0.90625	14.71753
0.953125	13.67044
1	12.72792
1.046875	11.87647
1.09375	11.10477
1.140625	10.40325
1.1875	9.763742
1.234375	9.179269
1.28125	8.643805
1.328125	8.152131
1.375	7.699702
1.421875	7.282548
1.46875	6.897183
1.515625	6.540537
1.5625	6.209899
1.609375	5.902863
1.65625	5.617294
1.703125	5.351288
1.75	5.103143
1.796875	4.871337
1.84375	4.6545
1.890625	4.451404
1.9375	4.260941
1.984375	4.082108
2.03125	3.914
2.078125	3.755796
2.125	3.606751
2.171875	3.466187
2.21875	3.333487
2.265625	3.208086
2.3125	3.089471
2.359375	2.97717
2.40625	2.870752
2.453125	2.769821
2.5	2.674013
2.546875	2.582994
2.59375	2.496455
2.640625	2.414113
2.6875	2.335705
2.734375	2.260988
2.78125	2.189739
2.828125	2.121748
2.875	2.056825
2.921875	1.994789
2.96875	1.935474
3.015625	1.878727
3.0625	1.824402
3.109375	1.772367
3.15625	1.722495
3.203125	1.67467
3.25	1.628783
3.296875	1.584732
3.34375	1.542422
3.390625	1.501763
3.4375	1.462671
3.484375	1.425068
3.53125	1.38888
3.578125	1.354037
3.625	1.320475
3.671875	1.288132
3.71875	1.25695
3.765625	1.226876
3.8125	1.197857
3.859375	1.169845
3.90625	1.142795
3.953125	1.116664
4	1.09141

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
Finite rod field	4.024922	N/C
Infinite-line approximation	9	N/C
Point-charge approximation	4.5	N/C
Infinite-line relative overestimate	1.236068	ratio
Point-charge relative overestimate	0.118034	ratio
Line density at fixed total charge	1	nC/m

A long-line limit fixes density, while a far-field point limit fixes total charge; those are different limiting comparisons.

Odd horizontal contributions cancel. Integrating Kλy ds/(s²+y²)^(3/2) gives 2KλL/[y√(L²+y²)]=KQ/[y√(L²+y²)]. Dividing each approximation by this exact expression gives √(1+(y/L)²) and √(1+(L/y)²). Neither approximation is uniformly good over all geometry.

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

A long-line limit fixes density, while a far-field point limit fixes total charge; those are different limiting comparisons.

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 vertical element is Kλy ds/(s²+y²)^(3/2). An antiderivative is Kλs/[y√(s²+y²)], so the exact field is 2KλL/[y√(L²+y²)]=KQ/[y√(L²+y²)]. The point comparison is KQ/y², and the infinite-line comparison is 2Kλ/y. Derive their ratios to the exact field rather than deciding from a diagram that one is close enough.

Check evidence and the domain

The point ratio is √(1+(L/y)²); the infinite-line ratio is √(1+(y/L)²). Both overestimate this finite-source result. A far-field limit keeps Q fixed, while increasing line length toward an infinite line ordinarily keeps λ fixed. The workbench fixes Q, so its changing λ is displayed explicitly. Approximation error is not a statistical uncertainty interval.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

At Q=2 nC, L=1 m and y=1 m, find the exact 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

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Previous: Normalize charge before comparing density moments Next: Test symmetry of the charge-weighted integrand

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