Differentiate ring potential without losing the reference distinction
Potential contributions add as scalars, which can make source integration simpler than vector field integration. Every element of a uniform ring is the same distance from an axial point. With Q=2 nC, V(z)=18/√(R²+z²) V when the reference is zero at infinity. This finite-source reference is legitimate because the potential tends to zero at large distance.
Learning outcome: Predict the center field from symmetry, then verify by differentiating the potential. At z=1 m compare h=0.1,0.2,0.4 m and retain signed errors. On paper add the same 5 V offset to both sampled potentials and show the cancellation before discussing independent error bounds.
Starting recap
Review oriented normals, definite integrals, source volume measures and partial derivatives as one-coordinate rates. Potential differences use an explicit reference and field work uses a signed line integral.
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For R=1 m, center potential is 18 V while center field is zero. At z=1 m, V=18/√2≈12.727922 V and exact E=9/√2≈6.363961 N/C. Compare symmetric half-spacings 0.1 and 0.2 m against that analytic derivative; their disagreement is discretization evidence, not a change in source charge.
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Which statement about the ring center is correct?
Experiment
Predict the center field from symmetry, then verify by differentiating the potential. At z=1 m compare h=0.1,0.2,0.4 m and retain signed errors. On paper add the same 5 V offset to both sampled potentials and show the cancellation before discussing independent error bounds.
Differentiate a source potential and audit a finite difference
Worked teaching model · synthetic values · Uniform ring Q=2 nC, potential V(z)=18/sqrt(R²+z²) V with infinity reference. Evaluate axial field at z=1 m by analytic derivative and symmetric finite difference. Added gauge offsets cancel in the difference; finite spacing gives truncation error, not a physical field change.
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Ring potential with infinity reference
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: source-integrated scalar potential (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Ring potential with infinity reference
source-integrated scalar potential
| axial coordinate (m) | potential (V) |
|---|---|
| -2 | 8.049845 |
| -1.95 | 8.2137 |
| -1.9 | 8.383436 |
| -1.85 | 8.559307 |
| -1.8 | 8.741573 |
| -1.75 | 8.930501 |
| -1.7 | 9.126362 |
| -1.65 | 9.329431 |
| -1.6 | 9.539981 |
| -1.55 | 9.758284 |
| -1.5 | 9.984604 |
| -1.45 | 10.21919 |
| -1.4 | 10.46229 |
| -1.35 | 10.7141 |
| -1.3 | 10.97479 |
| -1.25 | 11.24451 |
| -1.2 | 11.52332 |
| -1.15 | 11.81122 |
| -1.1 | 12.10811 |
| -1.05 | 12.41379 |
| -1 | 12.72792 |
| -0.95 | 13.04999 |
| -0.9 | 13.37929 |
| -0.85 | 13.71491 |
| -0.8 | 14.05564 |
| -0.75 | 14.4 |
| -0.7 | 14.74617 |
| -0.65 | 15.09199 |
| -0.6 | 15.43487 |
| -0.55 | 15.77189 |
| -0.5 | 16.09969 |
| -0.45 | 16.41459 |
| -0.4 | 16.71258 |
| -0.35 | 16.98945 |
| -0.3 | 17.24087 |
| -0.25 | 17.46257 |
| -0.2 | 17.65045 |
| -0.15 | 17.80085 |
| -0.1 | 17.91067 |
| -0.05 | 17.97754 |
| 0 | 18 |
| 0.05 | 17.97754 |
| 0.1 | 17.91067 |
| 0.15 | 17.80085 |
| 0.2 | 17.65045 |
| 0.25 | 17.46257 |
| 0.3 | 17.24087 |
| 0.35 | 16.98945 |
| 0.4 | 16.71258 |
| 0.45 | 16.41459 |
| 0.5 | 16.09969 |
| 0.55 | 15.77189 |
| 0.6 | 15.43487 |
| 0.65 | 15.09199 |
| 0.7 | 14.74617 |
| 0.75 | 14.4 |
| 0.8 | 14.05564 |
| 0.85 | 13.71491 |
| 0.9 | 13.37929 |
| 0.95 | 13.04999 |
| 1 | 12.72792 |
| 1.05 | 12.41379 |
| 1.1 | 12.10811 |
| 1.15 | 11.81122 |
| 1.2 | 11.52332 |
| 1.25 | 11.24451 |
| 1.3 | 10.97479 |
| 1.35 | 10.7141 |
| 1.4 | 10.46229 |
| 1.45 | 10.21919 |
| 1.5 | 9.984604 |
| 1.55 | 9.758284 |
| 1.6 | 9.539981 |
| 1.65 | 9.329431 |
| 1.7 | 9.126362 |
| 1.75 | 8.930501 |
| 1.8 | 8.741573 |
| 1.85 | 8.559307 |
| 1.9 | 8.383436 |
| 1.95 | 8.2137 |
| 2 | 8.049845 |
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 |
|---|---|---|
| Potential at ring center | 18 | V |
| Axial field at ring center | 0 | N/C |
| Potential at z equals 1 | 12.72792 | V |
| Analytic axial field at z equals 1 | 6.363961 | N/C |
| Symmetric finite-difference field | 6.330798 | N/C |
| Signed derivative approximation error | -0.03316259 | N/C |
A nonzero potential can have zero gradient; gauge offsets change neither field nor potential differences.
All ring elements lie at the same axial distance, so V=KQ/√(R²+z²) with zero at infinity. Taking −dV/dz gives KQz/(R²+z²)^(3/2). At the center V=KQ/R while E=0 by symmetry. A symmetric finite difference approximates the derivative at z=1; changing its spacing changes truncation error, not the underlying field. Adding any constant to both sampled potentials cancels exactly algebraically.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A nonzero potential can have zero gradient; gauge offsets change neither field nor potential differences.
All ring elements lie at the same axial distance, so V=KQ/√(R²+z²) with zero at infinity. Taking −dV/dz gives KQz/(R²+z²)^(3/2). At the center V=KQ/R while E=0 by symmetry. A symmetric finite difference approximates the derivative at z=1; changing its spacing changes truncation error, not the underlying field. Adding any constant to both sampled potentials cancels exactly algebraically.
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