SCIENCE · INVESTIGATE AND EXPLAIN

Derive a finite-line field and compare distinct limits

Use pairwise cancellation while retaining the full source distance.

Learning outcome: Compare half-lengths.5,1,2 m at fixed Q and distance 1 m. Derive the exact vertical field, then separately compare each with the infinite-line expression at that case’s actual density. Sketch paired horizontal contributions and retain their cancellation before integration.

Starting recap

Choose source and observer coordinates, integrate normalized charge, and justify symmetry before simplifying field or flux.

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Try

For a uniform line from−1 to 1 m with Q = 2 nC and observation at(0,1), lambda = 1 nC/m and Ey = kQ/sqrt 2≈12.7103 N/C. The same-density infinite-line expression gives 17.9751 N/C; the finite line has fewer contributing charges.

Open the experiment

Predict

Which quantity is held fixed when taking the infinite uniform line limit?

Your prediction and reason

Experiment

Compare half-lengths.5,1,2 m at fixed Q and distance 1 m. Derive the exact vertical field, then separately compare each with the infinite-line expression at that case’s actual density. Sketch paired horizontal contributions and retain their cancellation before integration.

Compare a finite line with its infinite limit

Worked teaching model · synthetic values · Uniform line spans −a to a at y=0; total charge fixed at 2 nC. Sample (0,b). Opposite horizontal components cancel by source symmetry. Infinite-line comparison holds the current lambda=Q/(2a), not Q fixed while length grows.

Source half-length: 1 m Perpendicular observation distance: 1 m

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Field versus perpendicular distance

Axes rescale independently. Curves join sampled coordinates; a joining segment is not additional measured evidence. Read physical units rather than screen angles or apparent slopes.

Field versus perpendicular distance: numerical coordinates

distance (m)	Ey (N/C)
0.25	69.7536
0.3125	54.902
0.375	44.8816
0.4375	37.6412
0.5	32.1548
0.5625	27.8518
0.625	24.3886
0.6875	21.5451
0.75	19.1734
0.8125	17.1701
0.875	15.4602
0.9375	13.9877
1	12.7103
1.0625	11.5948
1.125	10.6151
1.1875	9.75024
1.25	8.98317
1.3125	8.29995
1.375	7.68905
1.4375	7.14083
1.5	6.6472
1.5625	6.20131
1.625	5.79736
1.6875	5.43037
1.75	5.09609
1.8125	4.79082
1.875	4.5114
1.9375	4.25505
2	4.01936
2.0625	3.80221
2.125	3.60176
2.1875	3.41638
2.25	3.24461
2.3125	3.0852
2.375	2.937
2.4375	2.799
2.5	2.67031
2.5625	2.55013
2.625	2.43773
2.6875	2.33247
2.75	2.23377
2.8125	2.1411
2.875	2.05398
2.9375	1.97199
3	1.89474
3.0625	1.82188
3.125	1.75308
3.1875	1.68805
3.25	1.62653
3.3125	1.56827
3.375	1.51304
3.4375	1.46065
3.5	1.4109
3.5625	1.36362
3.625	1.31865
3.6875	1.27584
3.75	1.23507
3.8125	1.1962
3.875	1.15912
3.9375	1.12372
4	1.0899

Calculated values under the declared model
Quantity	Value	Unit
Line density	1	nC/m
Exact perpendicular field	12.7103	N/C
Midpoint field using 32 cells	12.7119	N/C
Infinite-line field at same density	17.9751	N/C
Point-charge far-field comparison	17.9751	N/C

Symmetry cancels horizontal components, not the denominators of vertical components.

Integrate k lambda y/(x²+y²)^(3/2). The long-line limit holds lambda fixed; the far-field point comparison holds totalQ fixed. Interchanging those constraints changes the question.

Derive before checking. Retain contrasting captures and explain one assumption that could fail; calculated evidence is not actual laboratory work.

Symmetry cancels horizontal components, not the denominators of vertical components.

Integrate k lambda y/(x²+y²)^(3/2). The long-line limit holds lambda fixed; the far-field point comparison holds totalQ fixed. Interchanging those constraints changes the question.

Observation to attach to a capture

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

Understand

Pair the source elements

A source at+x has an equal partner at−x. Their horizontal field components cancel at (0,y), while their vertical components add. Symmetry removes one component; it does not make the remaining source distance equal to y.

Integrate the remaining projection

For half-lengtha and total Q, lambda = Q/(2a). The vertical element is dEy = k lambda y dx/(x²+y²)^(3/2). Its integral over−a..a is Ey = kQ/[y sqrt(a²+y²)]. Each factor comes from projection and distance; using only inverse-square distance without projection would give a different answer.

Separate the two limits

For y much larger than a at fixed Q, Ey approaches kQ/y², the leading point-charge field. For a much larger than y at fixed lambda, Q = 2a lambda grows and Ey approaches 2k lambda/y. These are distinct limiting experiments. A plot made by holdingQ fixed cannot demonstrate the fixed-density infinite-line limit by itself.

Use the numerical comparator

The 32-cell midpoint result is independent of the antiderivative but shares the same source model. Compare its signed error against the exact expression; agreement checks computation, not the physical uniformity of an actual wire. A genuinely empirical claim would need separately recorded source and field evidence.

Worked original case

For a uniform line from−1 to 1 m with Q = 2 nC and observation at (0,1), lambda = 1 nC/m and Ey = kQ/sqrt 2≈12.7103 N/C. The same-density infinite-line expression gives 17.9751 N/C; the finite line has fewer contributing charges.

Investigation

Public numerical transfer practice

Try a new practice case

At distance 1 m and source half-lengthsqrt 3 m, what is Ey/[kQ/(1 m)²]?

Numerical result in the requested units

This public, unsaved check is not a secure assessment. Explain the source integral, symmetry or charge boundary in the retained course evidence workspace.

Sources and evidence boundary

Original cases and teaching checked 30 October 2026 against the current official calculus-based electricity and magnetism framework and OpenStax electric fields. All displayed records are generated or supplied, not learner measurements. No charging, mains connection, high-voltage equipment, physical circuit or material handling is requested. Public feedback and retained self-reports are practice evidence, not independent mastery, performed laboratory time or AP authorization.

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

Which source, boundary or independence assumption would have to change before the conclusion changes?

Your explanation and limitations

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Previous: Normalize a line density before integrating its field Next: Integrate ring projections before taking limits

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