Integrate an arc while total charge stays fixed
Parameterizing a curved source requires its length element. A circular arc of radius R at angle φ has ds=R dφ when φ is in radians. For a uniform arc from −θ to +θ carrying fixed charge Q, λ=Q/(2θR). Changing the angle while holding Q fixed therefore changes its line density.
Learning outcome: Compare half-angles 30°,90°,180° at fixed Q and radius. Retain the radian conversion, λ and center-field vector. Then double R on paper and explain the field scaling under fixed Q versus fixed λ.
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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For Q=2 nC, R=1 m and half-angle θ=90°=π/2, λ=2/π nC/m and Ex=−36/π≈−11.459156 N/C. Ey=0. At the same radius and fixed Q, a full ring gives exactly zero center field.
Open the experimentPredict
At fixed Q and arc angle, doubling R changes the center-field magnitude by which factor?
Experiment
Compare half-angles 30°,90°,180° at fixed Q and radius. Retain the radian conversion, λ and center-field vector. Then double R on paper and explain the field scaling under fixed Q versus fixed λ.
Integrate a curved source at fixed total charge
Worked teaching model · synthetic values · A uniform positive 2 nC arc centered at the origin lies at angles −θ to +θ around the positive x axis. Observe the center. Q is fixed as angle changes, so λ=Q/(2θR), with θ in radians. The complete 360-degree ring has exactly zero center field by symmetry.
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Original arc source geometry
Spatial axes use the same pixels per physical unit, preserving geometry. 1: charged arc, not field lines (solid).2: observation at center (open-circle marked location). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Original arc source geometry
charged arc, not field lines
| east coordinate (m) | north coordinate (m) |
|---|---|
| 0.5 | -0.8660254 |
| 0.5224986 | -0.8526402 |
| 0.544639 | -0.8386706 |
| 0.5664062 | -0.8241262 |
| 0.5877853 | -0.809017 |
| 0.6087614 | -0.7933533 |
| 0.6293204 | -0.777146 |
| 0.649448 | -0.760406 |
| 0.6691306 | -0.7431448 |
| 0.6883546 | -0.7253744 |
| 0.7071068 | -0.7071068 |
| 0.7253744 | -0.6883546 |
| 0.7431448 | -0.6691306 |
| 0.760406 | -0.649448 |
| 0.777146 | -0.6293204 |
| 0.7933533 | -0.6087614 |
| 0.809017 | -0.5877853 |
| 0.8241262 | -0.5664062 |
| 0.8386706 | -0.544639 |
| 0.8526402 | -0.5224986 |
| 0.8660254 | -0.5 |
| 0.8788171 | -0.4771588 |
| 0.8910065 | -0.4539905 |
| 0.9025853 | -0.4305111 |
| 0.9135455 | -0.4067366 |
| 0.9238795 | -0.3826834 |
| 0.9335804 | -0.3583679 |
| 0.9426415 | -0.3338069 |
| 0.9510565 | -0.309017 |
| 0.9588197 | -0.2840153 |
| 0.9659258 | -0.258819 |
| 0.9723699 | -0.2334454 |
| 0.9781476 | -0.2079117 |
| 0.9832549 | -0.1822355 |
| 0.9876883 | -0.1564345 |
| 0.9914449 | -0.1305262 |
| 0.9945219 | -0.1045285 |
| 0.9969173 | -0.0784591 |
| 0.9986295 | -0.05233596 |
| 0.9996573 | -0.02617695 |
| 1 | 0 |
| 0.9996573 | 0.02617695 |
| 0.9986295 | 0.05233596 |
| 0.9969173 | 0.0784591 |
| 0.9945219 | 0.1045285 |
| 0.9914449 | 0.1305262 |
| 0.9876883 | 0.1564345 |
| 0.9832549 | 0.1822355 |
| 0.9781476 | 0.2079117 |
| 0.9723699 | 0.2334454 |
| 0.9659258 | 0.258819 |
| 0.9588197 | 0.2840153 |
| 0.9510565 | 0.309017 |
| 0.9426415 | 0.3338069 |
| 0.9335804 | 0.3583679 |
| 0.9238795 | 0.3826834 |
| 0.9135455 | 0.4067366 |
| 0.9025853 | 0.4305111 |
| 0.8910065 | 0.4539905 |
| 0.8788171 | 0.4771588 |
| 0.8660254 | 0.5 |
| 0.8526402 | 0.5224986 |
| 0.8386706 | 0.544639 |
| 0.8241262 | 0.5664062 |
| 0.809017 | 0.5877853 |
| 0.7933533 | 0.6087614 |
| 0.777146 | 0.6293204 |
| 0.760406 | 0.649448 |
| 0.7431448 | 0.6691306 |
| 0.7253744 | 0.6883546 |
| 0.7071068 | 0.7071068 |
| 0.6883546 | 0.7253744 |
| 0.6691306 | 0.7431448 |
| 0.649448 | 0.760406 |
| 0.6293204 | 0.777146 |
| 0.6087614 | 0.7933533 |
| 0.5877853 | 0.809017 |
| 0.5664062 | 0.8241262 |
| 0.544639 | 0.8386706 |
| 0.5224986 | 0.8526402 |
| 0.5 | 0.8660254 |
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.
observation at center
| east coordinate (m) | north coordinate (m) |
|---|---|
| 0 | 0 |
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.
| Quantity | Value | Unit |
|---|---|---|
| Arc half-angle in radians | 1.047198 | rad |
| Line density required at fixed charge | 0.9549297 | nC/m |
| East field at center | -14.88588 | N/C |
| North field at center | 0 | N/C |
| Total source charge | 2 | nC |
The arc-length Jacobian and the fixed-charge normalization both depend on radius and angle.
dq=λR dφ and the center-point displacement reverses the radial unit vector. Hence Ex=−Kλ/R times integral cosφ dφ=−KQ sinθ/(R²θ), while Ey cancels. The θ→0 limit is a point source at (R,0); θ=π is a full ring with exactly zero field. Holding λ instead would change Q and answer a different question.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The arc-length Jacobian and the fixed-charge normalization both depend on radius and angle.
dq=λR dφ and the center-point displacement reverses the radial unit vector. Hence Ex=−Kλ/R times integral cosφ dφ=−KQ sinθ/(R²θ), while Ey cancels. The θ→0 limit is a point source at (R,0); θ=π is a full ring with exactly zero field. Holding λ instead would change Q and answer a different question.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
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