SCIENCE · INVESTIGATE AND EXPLAIN

Integrate an arc with signed component geometry

Retain angular limits, density normalization and two-component directions.

Learning outcome: Compare spanspi/4,pi/2 andpi at fixed total Q = 2 nC. Sketch the direction of each endpoint contribution, integrate both components and compare with the 16-cell sum. Then state how the comparison would change if line density, rather than total Q, were held fixed.

Starting recap

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

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Try

A positive 2 nC semicircle of radius 1 m from 0 topi gives Ex = 0 and Ey = −2kQ/pi≈−11.4433 N/C. The 16-cell midpoint sum has a small integration discrepancy; it is not a second physical observation.

Open the experiment

Predict

Which expression correctly accounts for charge on an angular element of a uniform arc?

Your prediction and reason

Experiment

Compare spanspi/4,pi/2 andpi at fixed total Q = 2 nC. Sketch the direction of each endpoint contribution, integrate both components and compare with the 16-cell sum. Then state how the comparison would change if line density, rather than total Q, were held fixed.

Integrate an arc as vector components

Worked teaching model · synthetic values · Uniform arc radius 1 m spans theta=0 to selected positive span ≤pi. Field at center points away from positive source elements toward center. Total charge held fixed for span comparisons; density is recomputed.

Arc angular span: 1 pi radians Arc total charge: 2 nC

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Differential x contribution

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.

Differential x contribution: numerical coordinates

source angle (rad)	dEx/dtheta (N/C)
0	-5.72165
0.0523599	-5.71381
0.10472	-5.69031
0.15708	-5.65121
0.20944	-5.59662
0.261799	-5.52669
0.314159	-5.44162
0.366519	-5.34162
0.418879	-5.22699
0.471239	-5.09803
0.523599	-4.9551
0.575959	-4.79858
0.628319	-4.62891
0.680678	-4.44656
0.733038	-4.25202
0.785398	-4.04582
0.837758	-3.82853
0.890118	-3.60075
0.942478	-3.3631
0.994838	-3.11624
1.0472	-2.86083
1.09956	-2.59758
1.15192	-2.32721
1.20428	-2.05046
1.25664	-1.76809
1.309	-1.48087
1.36136	-1.1896
1.41372	-0.895064
1.46608	-0.598076
1.51844	-0.299448
1.5708	-1.62081e-15
1.62316	0.299448
1.67552	0.598076
1.72788	0.895064
1.78024	1.1896
1.8326	1.48087
1.88496	1.76809
1.93732	2.05046
1.98968	2.32721
2.04204	2.59758
2.0944	2.86083
2.14675	3.11624
2.19911	3.3631
2.25147	3.60075
2.30383	3.82853
2.35619	4.04582
2.40855	4.25202
2.46091	4.44656
2.51327	4.62891
2.56563	4.79858
2.61799	4.9551
2.67035	5.09803
2.72271	5.22699
2.77507	5.34162
2.82743	5.44162
2.87979	5.52669
2.93215	5.59662
2.98451	5.65121
3.03687	5.69031
3.08923	5.71381
3.14159	5.72165

Differential y contribution

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.

Differential y contribution: numerical coordinates

source angle (rad)	dEy/dtheta (N/C)
0	0
0.0523599	-0.299448
0.10472	-0.598076
0.15708	-0.895064
0.20944	-1.1896
0.261799	-1.48087
0.314159	-1.76809
0.366519	-2.05046
0.418879	-2.32721
0.471239	-2.59758
0.523599	-2.86083
0.575959	-3.11624
0.628319	-3.3631
0.680678	-3.60075
0.733038	-3.82853
0.785398	-4.04582
0.837758	-4.25202
0.890118	-4.44656
0.942478	-4.62891
0.994838	-4.79858
1.0472	-4.9551
1.09956	-5.09803
1.15192	-5.22699
1.20428	-5.34162
1.25664	-5.44162
1.309	-5.52669
1.36136	-5.59662
1.41372	-5.65121
1.46608	-5.69031
1.51844	-5.71381
1.5708	-5.72165
1.62316	-5.71381
1.67552	-5.69031
1.72788	-5.65121
1.78024	-5.59662
1.8326	-5.52669
1.88496	-5.44162
1.93732	-5.34162
1.98968	-5.22699
2.04204	-5.09803
2.0944	-4.9551
2.14675	-4.79858
2.19911	-4.62891
2.25147	-4.44656
2.30383	-4.25202
2.35619	-4.04582
2.40855	-3.82853
2.46091	-3.60075
2.51327	-3.3631
2.56563	-3.11624
2.61799	-2.86083
2.67035	-2.59758
2.72271	-2.32721
2.77507	-2.05046
2.82743	-1.76809
2.87979	-1.48087
2.93215	-1.1896
2.98451	-0.895064
3.03687	-0.598076
3.08923	-0.299448
3.14159	-3.24162e-15

Calculated values under the declared model
Quantity	Value	Unit
Arc angular span	3.14159	rad
Line density	0.63662	nC/m
Center field x component	-7.00700e-16	N/C
Center field y component	-11.4433	N/C
Center field magnitude	11.4433	N/C
Sixteen-cell x estimate	-6.97574e-16	N/C
Sixteen-cell y estimate	-11.4617	N/C

Integrate both signed components; total charge is not a vector.

With dq=(Q/theta)dphi, center displacement is opposite each radial source position. A semicircle cancels Ex but retains downward Ey; a quarter arc has equal negative components.

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

Integrate both signed components; total charge is not a vector.

With dq=(Q/theta)dphi, center displacement is opposite each radial source position. A semicircle cancels Ex but retains downward Ey; a quarter arc has equal negative components.

Observation to attach to a capture

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

Understand

Resolve the source displacement

At source anglephi, a positive element lies at R(cosphi,sinphi). Its field at the center points along(−cosphi,−sinphi). The field magnitude of the element is k dq/R². A diagram showing arrows away from the center would describe the field at the sources from a center charge, not this question.

Normalize the angular density

For a uniform arc spanning 0 totheta, line density is lambda = Q/(Rtheta) and ds = R dphi. Thereforedq = (Q/theta)dphi. The cancellation of R in this charge element does not remove the inverse-square factor from the electric field. Use radians when evaluating the angular integral.

Integrate both components

Integrating yields Ex = −kQ sin(theta)/(R²theta) and Ey = −kQ[1−cos(theta)]/(R²theta). A semicircle has zeroEx by reflection symmetry and negativeEy. A quarter-circle has equal negative components. A shorter arc at fixed Q has larger density, so it is not the same intervention as simply removing charge from a fixed-density arc.

Keep numerical residuals in context

Finite precision can display a tiny nonzeroEx for theta = pi even though the exact sine is zero. A midpoint sum also approximates the integral. Identify which discrepancy comes from floating-point trigonometry and which from finite cells; neither establishes broken physical symmetry in a generated packet.

Worked original case

A positive 2 nC semicircle of radius 1 m from 0 to pi gives Ex = 0 and Ey = −2kQ/pi≈−11.4433 N/C. The 16-cell midpoint sum has a small integration discrepancy; it is not a second physical observation.

Investigation

Public numerical transfer practice

Try a new practice case

For a uniform positive quarter-circle from 0 topi/2, what is |Ey|/|Ex| at its center?

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.

Apply

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: Integrate ring projections before taking limits Next: Reconcile six signed fluxes without guessing a local field

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