Distinguish radius-ratio invariance from field scale
For a very long coaxial pair, a Gaussian cylinder gives E=λ/(2πε0r)=2Kλ/r in the gap. Integrate radially from inner radius a to outer radius b to obtain ΔV=2Kλln(b/a). The logarithm requires a dimensionless radius ratio; a lone logarithm of a measured length would hide a reference.
Learning outcome: Compare a=0.005,0.01,0.02 m at fixed ratio 3. Retain capacitance per length and inner field separately. Then vary only the ratio and compare the derivative sensitivity with a small finite change, declaring which geometric quantity is held fixed.
Starting recap
Review charge conservation, conductor equipotential conditions, definite field integrals and energy derivatives under a stated electrical constraint. Convert capacitance and charge units before interpreting small energies.
All lessons are open directly. Your saved experiments remain when you leave and return.
Try
For a=0.01 m, b/a=3 and V=6 V, Cprime=1/[18×10⁹ ln3]≈50.568847 pF/m. The inner gap-side field is 600/ln3≈546.143536 N/C. Doubling both radii leaves Cprime fixed but halves that surface field.
Open the experimentPredict
At fixed radius ratio and voltage, doubling both coaxial radii gives:
Experiment
Compare a=0.005,0.01,0.02 m at fixed ratio 3. Retain capacitance per length and inner field separately. Then vary only the ratio and compare the derivative sensitivity with a small finite change, declaring which geometric quantity is held fixed.
Derive capacitance per length from a logarithmic voltage
Worked teaching model · synthetic values · Ideal very long vacuum coaxial conductors, outer radius b times inner radius a, voltage6 V. Report per-length capacitance and a hypothetical .5 m segment with end effects neglected. Outer-radius sensitivity varies outer radius at fixed inner radius; common geometric scaling is a different change.
Reset keeps your prediction, captures and practice history.
Coaxial field between conductor surfaces
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: voltage-normalized inverse-radius field (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Coaxial field between conductor surfaces
voltage-normalized inverse-radius field
| radius (m) | radial field (N/C) |
|---|---|
| 0.01 | 546.1435 |
| 0.01025 | 532.823 |
| 0.0105 | 520.1367 |
| 0.01075 | 508.0405 |
| 0.011 | 496.4941 |
| 0.01125 | 485.4609 |
| 0.0115 | 474.9074 |
| 0.01175 | 464.803 |
| 0.012 | 455.1196 |
| 0.01225 | 445.8315 |
| 0.0125 | 436.9148 |
| 0.01275 | 428.3479 |
| 0.013 | 420.1104 |
| 0.01325 | 412.1838 |
| 0.0135 | 404.5508 |
| 0.01375 | 397.1953 |
| 0.014 | 390.1025 |
| 0.01425 | 383.2586 |
| 0.0145 | 376.6507 |
| 0.01475 | 370.2668 |
| 0.015 | 364.0957 |
| 0.01525 | 358.1269 |
| 0.0155 | 352.3507 |
| 0.01575 | 346.7578 |
| 0.016 | 341.3397 |
| 0.01625 | 336.0883 |
| 0.0165 | 330.9961 |
| 0.01675 | 326.0558 |
| 0.017 | 321.2609 |
| 0.01725 | 316.6049 |
| 0.0175 | 312.082 |
| 0.01775 | 307.6865 |
| 0.018 | 303.4131 |
| 0.01825 | 299.2567 |
| 0.0185 | 295.2127 |
| 0.01875 | 291.2766 |
| 0.019 | 287.444 |
| 0.01925 | 283.7109 |
| 0.0195 | 280.0736 |
| 0.01975 | 276.5284 |
| 0.02 | 273.0718 |
| 0.02025 | 269.7005 |
| 0.0205 | 266.4115 |
| 0.02075 | 263.2017 |
| 0.021 | 260.0684 |
| 0.02125 | 257.0087 |
| 0.0215 | 254.0202 |
| 0.02175 | 251.1005 |
| 0.022 | 248.2471 |
| 0.02225 | 245.4578 |
| 0.0225 | 242.7305 |
| 0.02275 | 240.0631 |
| 0.023 | 237.4537 |
| 0.02325 | 234.9004 |
| 0.0235 | 232.4015 |
| 0.02375 | 229.9552 |
| 0.024 | 227.5598 |
| 0.02425 | 225.2138 |
| 0.0245 | 222.9157 |
| 0.02475 | 220.6641 |
| 0.025 | 218.4574 |
| 0.02525 | 216.2945 |
| 0.0255 | 214.1739 |
| 0.02575 | 212.0946 |
| 0.026 | 210.0552 |
| 0.02625 | 208.0547 |
| 0.0265 | 206.0919 |
| 0.02675 | 204.1658 |
| 0.027 | 202.2754 |
| 0.02725 | 200.4196 |
| 0.0275 | 198.5976 |
| 0.02775 | 196.8085 |
| 0.028 | 195.0513 |
| 0.02825 | 193.3251 |
| 0.0285 | 191.6293 |
| 0.02875 | 189.963 |
| 0.029 | 188.3254 |
| 0.02925 | 186.7157 |
| 0.0295 | 185.1334 |
| 0.02975 | 183.5777 |
| 0.03 | 182.0478 |
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 |
|---|---|---|
| Outer radius | 0.03 | m |
| Capacitance per unit length | 5.05688e-11 | F/m |
| Capacitance of .5 m ideal segment | 2.52844e-11 | F |
| Charge per unit length at 6 V | 3.03413e-10 | C/m |
| Inner gap-side field | 546.1435 | N/C |
| Logarithmic sensitivity to outer radius at fixed inner radius | -0.9102392 | ratio |
| Stored energy per length | 9.10239e-10 | J/m |
Coaxial capacitance per length depends on the radius ratio, while the surface field also depends on absolute scale.
Gauss gives E=lambda/(2πepsilon0 r)=2Klambda/r. Integrating between inner and outer radii gives DeltaV=2Klambda ln(bouter/ainner), so Cprime=1/[2K ln(ratio)]. Scaling both radii together leaves Cprime unchanged but reduces the field at corresponding scaled positions. Varying only the outer radius gives d ln Cprime/d ln bouter=−1/ln(ratio). Finite ends would require a fringe correction.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Coaxial capacitance per length depends on the radius ratio, while the surface field also depends on absolute scale.
Gauss gives E=lambda/(2πepsilon0 r)=2Klambda/r. Integrating between inner and outer radii gives DeltaV=2Klambda ln(bouter/ainner), so Cprime=1/[2K ln(ratio)]. Scaling both radii together leaves Cprime unchanged but reduces the field at corresponding scaled positions. Varying only the outer radius gives d ln Cprime/d ln bouter=−1/ln(ratio). Finite ends would require a fringe correction.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
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?
Continue
Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.
Skipping retains experiments and awards no mastery.
Save a science backup or import one
Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.
A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.
Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.
Complete file sets and course backups
A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.
The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.
0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.
Science storage: loading. No account or remote computation is required.